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Global Algorithm/Formulas For The Zeros Of Riemann's Zeta Function

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It is my belief that RH is a genuinely arithmetic question that likely will not succumb to methods of analysis. Number theorists are on the right track to an eventual proof of RH, but we are still lacking many of the tools. 

J. Brian Conrey

"...the Riemann Hypothesis will be settled without any fundamental changes in our mathematical thoughts, namely, all tools are ready to attack it but just a penetrating idea is missing." 
 
Y. Motohashi

"...there have been very few attempts at proving the Riemann hypothesis, because, simply, no one has ever had any really good idea for how to go about it." 

A. Selberg

"I still think that some major new idea is needed here" 

E. Bombieri

riem.jpg


http://wwwf.imperial.ac.uk/~hjjens/Riemann_talk.pdf

Subtle relations: prime numbers, complex functions, energy levels and Riemann

Prof. Henrik J. Jensen, Department of Mathematics, Imperial College London

http://www.ejtp.com/articles/ejtpv10i28p111.pdf

Riemann Zeta Function and Hydrogen Spectrum

The year: 1972. The scene: Afternoon tea in Fuld Hall at the Institute for Advanced Study. The camera pans around the Common Room, passing by several Princetonians in tweeds and corduroys, then zooms in on Hugh Montgomery, boyish Midwestern number theorist with sideburns. He has just been introduced to Freeman Dyson, dapper British physicist.

Dyson: So tell me, Montgomery, what have you been up to? 
Montgomery: Well, lately I've been looking into the distribution of the zeros of the Riemann zeta function. 
Dyson: Yes? And? 
Montgomery: It seems the two-point correlations go as.... (turning to write on a nearby blackboard):

20036410348_150.gif

Dyson: Extraordinary! Do you realize that's the pair-correlation function for the eigenvalues of a random Hermitian matrix? It's also a model of the energy levels in a heavy nucleus—say U-238.

 

The asymptotic formula developed by Riemann (discovered by C. Siegel in the early 1930s from the notes left by Riemann) is the most difficult asymptotic expansion ever attempted, certainly the most complex calculation of the 19th century. C. Siegel realized that no one else could have done it, and in 1930 Riemann was still ahead of every other mathematician involved in the study of the zeta function.


https://michaelberryphysics.files.wordpress.com/2013/06/berry483.pdf
 

saddle.jpg

 

saddle2.jpg

 

Why would B.F. Riemann embark on such a colossal derivation of an asymptotic expansion (see  H.M. Edwards, Riemann's Zeta Function, chapter 7) unless he was certain that all of the zeros do lie on the critical 1/2 line? The notes discovered by Siegel baffled the mathematicians, because Riemann used this most difficult asymptotic formula to simply obtain the values of the first few zeros of the zeta function. It is as if he had already proven that all of the zeros lie on the critical 1/2 line and he wanted to make sure that just the very first zeros have this property.

In my opinion, Riemann must have used both his newly discovered zeta functional equation and other equations in the Nachlass to prove the RH. Then, and only then, did he embark on this very difficult derivation.

 

Let us now briefly explore the best papers published on the RH.

Two mathematicians from the Lomonosov Moscow State University have used the mollifier function introduced by N. Levinson in a novel way:

https://arxiv.org/pdf/1805.07741.pdf

100% OF THE ZEROS OF THE RIEMANN ZETA-FUNCTION ARE ON THE CRITICAL LINE

Earlier, they published another paper in which they showed that at least 47% of the zeros of the Riemann zeta function lie on the critical line (the previous records were Feng (41%), Conrey (40%) and Levinson (34%)).

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.758.4457&rep=rep1&type=pdf

https://arxiv.org/pdf/1403.5786.pdf (the original paper on the novel way of using mollifier functions)

https://arxiv.org/pdf/1207.6583.pdf

Limitations to mollifying ζ(s)

http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=F7C33227D1D6635FFBC27972BA54E5A8?doi=10.1.1.36.9777&rep=rep1&type=pdf

Long mollifiers of the Riemann zeta function

https://arxiv.org/pdf/1604.02740.pdf

THE θ = ∞ CONJECTURE IMPLIES THE RIEMANN HYPOTHESIS

https://rjlipton.wordpress.com/2018/09/26/reading-into-atiyahs-proof/ (on Sir M. Atiyah's use of the Todd function for the Riemann hypothesis)

Mathematicians complain that 99% of the proofs submitted to the Annals are rejected because they make use of the zeta functional equation.

"The reason for this is (as has been known since the work of Davenport and Heilbronn) that there are many examples of zeta-like functions (e.g., linear combinations of L-functions) which enjoy a functional equation and similar analyticity and growth properties to zeta, but which have zeroes off of the critical line. Thus, any proof of RH must somehow use a property of zeta which has no usable analogue for the Davenport-Heilbronn examples."

However, the arguments used in the following papers are very well presented and make a lot of sense.

Riemann's nachlass = manuscripts, lecture notes, calculation sheets and letters left by G.F.B Riemann

https://www.researchgate.net/publication/281403728_To_unveil_the_truth_of_the_zeta_function_in_Riemann_Nachlass

The authors assert that not all of the formulas left by Riemann in his notes have been taken into consideration, and that these neglected equations were used by Riemann to actually prove the RH.


https://arxiv.org/ftp/arxiv/papers/0801/0801.4072.pdf

A Necessary Condition for the Existence of the Nontrivial Zeros of the Riemann Zeta Function

(a paper which shows that B. Riemann must have followed a similar kind of argument, using the newly discovered zeta functional equation, to reach the conclusion that all the nontrivial zeros are all located on the ½ line)

 

On the computation of the zeta zeros

The complexity of the Riemann-Siegel coefficients:

riem3.jpg
 

The Riemann-Siegel formula does not deal with the distribution of zeros.

Nor can it reveal the hidden pattern/structure of the zeta zeros.

That is why for very large values of the zeta zeros, the Euler-Maclaurin formula becomes competitive.

An alternative to the Riemann-Siegel formula: improving the convergence of the Euler-Maclaurin expansion thereby greatly reducing the length of the main sum:

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.758.2810&rep=rep1&type=pdf

 

Recently, A. LeClair and G. França introduced and proved a new formula for finding the values of the zeta zeros, which is faster than the Riemann-Siegel formula:

Transcendental equations satisfied by the individual zeros of
Riemann ζ, Dirichlet and modular L-functions

https://arxiv.org/pdf/1502.06003.pdf

Statistical and other properties of Riemann zeros based on an
explicit equation for the n-th zero on the critical line

https://arxiv.org/pdf/1307.8395.pdf

 

(to be continued; in the subsequent messages I will introduce the new global algorithm which only uses simple arithmetical operations to find the zeros of the zeta function, global formulas for the Lehmer pairs and large gaps, and other results)

De Bruijn-Newman constant

leh1.jpg


https://arxiv.org/pdf/1508.05870.pdf

Lehmer pairs revisited

The Riemann hypothesis means that the de Bruijn-Newman constant is zero.

Unusually close pairs of zeros of the Riemann zeta function, the Lehmer pairs, can be used to give lower bounds on Λ.

Soundararajan’s Conjecture B implies the existence of infinitely many strong Lehmer pairs, and thus, that the de Bruijn-Newman constant Λ is 0.


http://www.math.kent.edu/~varga/pub/paper_209.pdf

Lehmer pairs of zeros and the Riemann ξ-function

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.30.9492&rep=rep1&type=pdf

A new Lehmer pair of zeros and a new lower bound for the de Bruijn-Newman constant Λ

http://www.academia.edu/19018042/Lehmer_pairs_of_zeros_the_de_Bruijn-Newman_constant_and_the_Riemann_Hypothesis

Lehmer pairs of zeros, the de Bruijn-Newman constant Λ, and the Riemann Hypothesis

http://www.dtc.umn.edu/~odlyzko/doc/debruijn.newman.pdf 

An improved bound for the de Bruijn-Newman constant

https://www.ams.org/journals/mcom/2011-80-276/S0025-5718-2011-02472-5/S0025-5718-2011-02472-5.pdf

An improved lower bound for the de Bruijn-Newman constant


Recently, it was proven that the de Bruijn-Newman constant is non-negative:

https://arxiv.org/pdf/1801.05914.pdf

This means that an infinite sequence of Lehmer pairs of arbitrarily high quality (strong Lehmer pairs) will prove that the de Bruijn-Newman constant is equal to zero (Λ = 0).

https://terrytao.wordpress.com/2018/01/20/lehmer-pairs-and-gue/

 

If the de Bruijn-Newman constant is equal to zero, Λ = 0, then Riemann's hypothesis (all zeta zeros lie on the 1/2 critical line) is true.

However, in order to prove that -10-20 < Λ, at least 1030 zeros would have to be examined. The total number of simple arithmetic mathematical operations that have been performed by all digital computers in history is only on the order of 1023.

Not even with improvements in hardware, it cannot be hoped to compute 1030 zeta zeros using existing methods.

Strong/high quality Lehmer pairs can be used to give lower bounds for Λ.

The existence of infinitely many Lehmer pairs implies that the de Bruijn-Newman constant Λ is equal to 0.

 

Strong Lehmer pairs

1187 pairs:

http://www.slideshare.net/MatthewKehoe1/riemanntex (pg. 64-87)

riem2.jpg

leh4.jpg

highzeta.jpg

 

Very interesting comments on the S(t) function:

https://arxiv.org/pdf/1407.4358.pdf (page 46)

 


http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.751.9485&rep=rep1&type=pdf

"The RH and (5.3) imply that, as t → ∞, the graph of Z(t) will consist of tightly packed spikes, which will be more and more condensed as t increases, with larger and large oscillations. This I find hardly conceivable. Of course, it could happen that the RH is true and that (5.3) is not."

NEW COMPUTATIONS OF THE RIEMANN ZETA FUNCTION ON THE CRITICAL LINE

https://arxiv.org/pdf/1607.00709.pdf

As a byproduct of our search for large values, we also find large values of S(t). It is always the case in our computations that when ζ(1/2 + it) is very large there is a large gap between the zeros around the large value. And it seems that to compensate for this large gap the zeros nearby get “pushed” to the left and right. A typical trend in the large values that we have found is that S(t) is particularly large and positive before the large value and large and negative afterwards.

The calculations involve more than 50000 zeros in over 200 small intervals going up to the 1036th zero.

S(t) is related to the large gaps between the zeta zeros where high extreme values of peaks occur, where it seems to protect the zeta function from attaining the tightly packed spikes conjectured by mathematicians.

 


http://www.dhushara.com/DarkHeart/RH2/RH.htm (one of the very best works on the Riemann zeta function and the RH)

http://www.math.sjsu.edu/~goldston/Tsang%20Ch2.pdf (on the function S(t))

https://www.sciencedirect.com/science/article/pii/0022314X8790059X

https://math.boku.ac.at/udt/vol10/no2/09OzSteu.pdf (on the distribution of the argument of the zeta function)

http://wayback.cecm.sfu.ca/~pborwein/TEMP_PROTECTED/book.pdf (a classic work on the Riemann zeta function, it includes all of the major papers published over the last 150 years on the subject)

http://www.dtc.umn.edu/~odlyzko/unpublished/zeta.10to20.1992.pdf (one of the best papers on the zeta function, it includes pertinent material on the S(t) function, pg. 11, 25, 29, 43, 68)

 

N(T) = T/2π (logT/2π - 1) + 7/8 + o(1) + Nosc(T)

Nosc(T) = S(T) = 1/π Im log ζ(1/2 + iT), the oscillatory part of the formula 

<N(T)> = N(T) - Nosc(T)

"We have all this evidence that the Riemann zeros are vibrations, but we don't know what's doing the vibrating." 

"Maybe we have become so hung up on looking at the primes from Gauss's and Riemann's perspective that what we are missing is simply a different way to understand these enigmatic numbers. Gauss gave an estimate for the number of primes, Riemann predicted that the guess is at worst the square root of N off its mark, Littlewood showed that you can't do better than this. Maybe there is an alternative viewpoint that no one has found because we have become so culturally attached to the house that Gauss built." 

M. du Sautoy, The Music of the Primes

"The zeta function is probably the most challenging and mysterious object of modern mathematics, in spite of its utter simplicity. . . The main interest comes from trying to improve the Prime Number Theorem, i.e. getting better estimates for the distribution of the prime numbers. The secret to the success is assumed to lie in proving a conjecture which Riemann stated in 1859 without much fanfare, and whose proof has since then become the single most desirable achievement for a mathematician."

M.C. Gutzwiller, Chaos in Classical and Quantum Mechanics, page 308

"Riemann showed the importance of study of [the zeta] function for a range of problems in number theory centering around the distribution of prime numbers, and he further demonstrated that many of these problems could be settled if one knew the location of the zeros of this function. In spite of continued assaults and much progress since Riemann's initial investigations this tantalizing question remains one of the major unsolved problems in mathematics."

D. Reed, Figures of Thought (Routledge, New York, 1995) p.123

...it is that incidental remark - the Riemann Hypothesis - that is the truly astonishing legacy of his 1859 paper. Because Riemann was able to see beyond the pattern of the primes to discern traces of something mysterious and mathematically elegant at work - subtle variations in the distribution of those prime numbers. Brilliant for its clarity, astounding for its potential consequences, the Hypothesis took on enormous importance in mathematics. Indeed, the successful solution to this puzzle would herald a revolution in prime number theory. Proving or disproving it became the greatest challenge of the age...

It has become clear that the Riemann Hypothesis, whose resolution seems to hang tantalizingly just beyond our grasp holds the key to a variety of scientific and mathematical investigations. The making and breaking of modern codes, which depend on the properties of the prime numbers, have roots in the Hypothesis. In a series of extraordinary developments during the 1970s, it emerged that even the physics of the atomic nucleus is connected in ways not yet fully understood to this strange conundrum. ...Hunting down the solution to the Riemann Hypothesis has become an obsession for many - the veritable 'great white whale' of mathematical research. Yet despite determined efforts by generations of mathematicians, the Riemann Hypothesis defies resolution.""

J. Derbyshire, from the dustjacket description of Prime Obsession (John Henry Press, 2003)

"Proving the Riemann hypothesis won't end the story. It will prompt a sequence of even harder, more penetrating questions. Why do the primes achieve such a delicate balance between randomness and order? And if their patterns do encode the behaviour of quantum chaotic systems, what other jewels will we uncover when we dig deeper?

Those who believe mathematics holds the key to the Universe might do well to ponder a question that goes back to the ancients: What secrets are locked within the primes?"

E. Klarreich, "Prime Time" (New Scientist, 11/11/00)

"Riemann's insight followed his discovery of a mathematical looking-glass through which he could gaze at the primes. Alice's world was turned upside down when she stepped through her looking-glass. In contrast, in the strange mathematical world beyond Riemann's glass, the chaos of the primes seemed to be transformed into an ordered pattern as strong as any mathematician could hope for. He conjectured that this order would be maintained however far one stared into the never-ending world beyond the glass. His prediction of an inner harmony on the far side of the mirror would explain why outwardly the primes look so chaotic. The metamorphosis provided by Riemann's mirror, where chaos turns to order, is one which most mathematicians find almost miraculous. The challenge that Riemann left the mathematical world was to prove that the order he thought he could discern was really there." 

"For centuries, mathematicians had been listening to the primes and hearing only disorganised noise. These numbers were like random notes wildly dotted on a mathematical stave with no discernible tune. Now Riemann had found new ears with which to listen to these mysterious tones. The sine-like waves that Riemann had created from the zeros in his zeta landscape revealed some hidden harmonic structure." 

"These zeros did not appear to be scattered at random. Riemann's calculations indicated that they were lining up as if along some mystical ley line running through the landscape." 

 

(to be continued)

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Global algorithm for the zeros of the zeta function

The Riemann Zeta function takes the prize for the most complicated and enigmatic function.

The lack of a proof of the Riemann hypothesis doesn’t just mean we don’t know all the zeros are on the line x = 1/2, it means that, despite all the zeros we know of lying neatly and precisely smack bang on the line x = 1/2, no one knows why any of them do, for if we had a definitive reason why the first zero 1/2 + 14.13472514i has real value precisely 1/2, we would have a definitive reason to know why they all do.

Since the energy levels of the atoms are reflected in the zeros of the Riemann zeta landscape (see the calculation of the moments which give rise to a sequence of numbers, 1,  42,  24,042... verified by the quantum physicists and by the mathematicians), there must exist a certain interval which best captures all of the important features/formulas of the Riemann zeta function.

The value of the first zero is 14.134725...

List of zeroes of the zeta function:

http://www.dtc.umn.edu/~odlyzko/zeta_tables/index.html
 

14.134725 x 45 = 636.062625...

200/π  = 63.661977...

1400/22 = 63.636363...

The interval which best captures/describes the zeta function is 63.63636363...

Now, here is the most crucial observation.

zeros.jpg

Virtually all of the mathematicians have forgotten that there are TWO zeta functions to be investigated: only the zeros on the critical 1/2 line whose imaginary parts are positive have been researched and the corresponding zeta function, while the zeros on the critical 1/2 line whose imaginary parts are negative (and the corresponding zeta function) have been left aside, rarely even mentioned in most papers.

In order to reach any kind of meaningful results, especially using ONLY the tools provided by basic arithmetic, we must use BOTH these zeta functions in a novel way.

That is, we must translate the second zeta function to that interval: now we will have two zeta functions within the same interval.

"The zeta function is probably the most challenging and mysterious object of modern mathematics, in spite of its utter simplicity."

"We may – paraphrasing the famous sentence of George Orwell – say that 'all mathematics is beautiful, yet some is more beautiful than the other'. But the most beautiful in all mathematics is the zeta function. There is no doubt about it."

The basic interval of 63.63636363 will be divided like a fractal, according to these values/proportions:

26.7
53.4
80
136.1
534

Applying these five elements proportions to the our fundamental distance, we get:

63.636363
16.1773
9.5445
6.36363
3.1815

That is, 534 is divided by 20 to obtain 26.7, and divided by 10 to get 53.4, again divided by approximately 20/3 to get 80, and 534/136.1 =~ 1/(4 x 0.063636363...).

We divide the 63.636363 interval in the same way (same proportional values) to obtain: 63.636363, 16.1773, 9.5445, 6.3636363, 3.1815.

From the left to the right, the first the zeta function will include the zeros from 14.134725 to 77.771.

From right to left, the second  translated zeta function will include the zeros from 14.134725 to 77.771 pointing the other way.

77.771... - 14.134725 = 63.63636363...

Here are the values of the zeta zeros on the first interval (from 14.134725... to 77.771 over a distance of 63.6363636... units):

     14.134725142
     21.022039639
     25.010857580
     30.424876126
     32.935061588
     37.586178159
     40.918719012
     43.327073281
     48.005150881
     49.773832478
     52.970321478
     56.446247697
     59.347044003
     60.831778525
     65.112544048
     67.079810529
     69.546401711
     72.067157674
     75.704690699
     77.144840069

Now, we will further subdivide (in a way the basic interval will become a fractal) each subsequent interval according to the same values/proportions, as follows:

63.636363
16.1773
9.5445
6.36363
3.1815

9.5445 - 6.36363 = 6.36363 - 3.1815 = 3.1815

3.1815
0.80886
0.477225
0.31815
0.159075

16.1773 - 9.5445 = 6.6328

6.6328
1.68632
0.99492
0.66328
0.33164


63.6363 - 16.1773 = 47.459

47.459
12.066
7.11885
4.7459
2.373

47.459 - 12.066 = 35.393

35.393
8.998
5.309
3.5393
1.77

35.393 - 8.998 = 26.395

26.395
6.7106
3.96
2.6395
1.31975

26.395 - 6.7106 = 19.6894

19.6894 
5.0045
2.95266
1.96894
0.98422

19.6894 - 5.0045 = 14.68

14.68 
3.7372
2.202
1.468
0.734

14.68 - 3.7372 = 10.9478

10.9478
2.7834
1.6422
1.09478
0.5474

10.9478 - 2.7834 = 8.1694

8.1694 
2.0757
1.22466
0.81694
0.40822

8.1694 - 2.0757 = 6.0887

6.0887
1.548
0.9133
0.60487
0.304425

6.0887 - 1.548 = 4.5407

4.5407
1.1544
0.681105
0.45407
0.227035

4.5407 - 1.1544 = 3.3863

3.3863
0.861
0.508
0.33863
0.169315

3.3863 - 0.861 = 2.5253

2.5253
0.692
0.3788
0.25253
0.126

2.5253 - 0.692 = 1.833

1.833
0.4661
0.275
0.1833
0.09


12.066 - 7.11885 = 4.947

4.947
1.2577
0.74205
0.4947
0.28735
 

Then, the values of the subdivision of our basic interval using the five elements ratios/proportions, will nearly coincide with the values of the zeroes of Riemann's zeta function.

14.134725142  
     21.022039639  20.497725 (14.134725 + 6.363)
     25.010857580  23.679225 (14.134725 + 9.5445)
     30.424876126  30.312025 (14.134725 + 16.1773)
     32.935061588  32.685 (30.312025 + 2.373)
     37.586178159  37.43 (30.312 + 7.11885)
     40.918719012  40.5234 (77.7647 - 16.1773 - 12.066 - 8.998)
     43.327073281  42.37 (30.312 + 12.066)
     48.005150881  47.68 (42.37 + 5.309)
     49.773832478  49.554 (77.7647 - 16.1773 - 12.066)
     52.970321478  51.37 (42.37 + 8.998)
     56.446247697  55.336 (51.37 + 3.96)
     59.347044003  58.08 (51.37 + 6.7106) or 59.07 (51.37 + 6.7106 + 0.98422)
     60.831778525  60.05 (58.08 + 1.96844)
     65.112544048  65.06 (60.05 + 5.0045)
     67.079810529  67.26 (65.06 + 2.202)
     69.546401711  68.79 (65.06 + 3.7322)
     72.067157674  71.575 (68.79 + 2.7834)
     75.704690699  75.1988 (71.575 + 2.0757 + 1.548)
     77.144840069  77.214 (75.1988 + 1.1544 + 0.861)
 

To put it another way:

For the first zeta function, the subdivision points will look like this:

14.1347 +

3.1815  = 17.3162 *
6.363    = 20.4947 *
9.545    = 23.68 *
16.1773 = 30.312 *
2.373    = 32.685 *
4.746    = 35.058
7.1185  = 37.43 } *
12.066  = 42.378 }
1.77      = 44.148 *
3.54     = 45.92 * midpoint
5.309   = 47.687 *
{8.998   = 51.376
{1.319   = 52.695
2.64     = 54.02 *
3.96     = 55.33
6.7106 = 58.086 *
0.984  = 59.07 *
1.968  = 60.05
2.95   = 61.03 *
5.0045 = 63.1 }
0.734  = 63.8 }
1.468  = 64.56 }
2.2     = 65.3 }
3.73   = 66.8 }
1.64   = 68.46 }
2.783  = 69.6
1.224  = 70.83
2.07   = 71.67 *
1.548  = 73.22
1.154  = 74.38 *
0.861  = 75.24
0.692  = 75.93
0.4661 = 76.4
0.3475 = 76.745
0.26   = 77

 

For the second zeta function, right on the same interval, we will obtain:

 

14.1347 + 63.63 = 77.7647

77.7647 -

3.1815 = 74.58 *
6.363  = 71.4 *
9.545  = 68.22 * }
16.173 = 61.587 * }
2.373  = 59.2 *
4.746  = 56.84 *
7.1185 = 54.469 * }
12.066 = 49.52 }
1.77  = 47.75 *
3.54 = 45.98 * midpoint
5.309 = 44.2 *
8.998 = 40.52 }
1.319 = 39.2 }
2.64  = 37.88 }
3.96  = 36.56 *
6.7106 = 33.8
0.984  = 32.8 *
1.968  = 31.84
2.95  = 30.86 *
5.0045 = 28.81 
0.734 = 28.07
1.468 = 27.34
2.2 = 26.61
3.73 = 25.08
1.64 = 23.43 *
2.783 = 22.23
1.224  = 21.07
2.07  = 20.22 *
1.548 = 18.67
1.154 = 17.52 * 
0.861 = 16.66
0.692 = 15.97
0.4661 = 15.5 
0.3475 = 15.15 
0.26 = 14.9

2π/log(t/2π) is the average gap/spacing formula.

 

Now, we are ready for the global algorithm for the zeta zeros.

 

z1 = 14.134725

L(z1) = 7.74977

14.134725 + 7.74977 = 21.884497

Now, all of the previous results will be applied to obtain the value of the second zero of the zeta function, to four decimal places accuracy, using only the five elements subdivision applied to both zeta functions as a guide.

63.636363
16.1773
9.5445
6.36363
3.1815

9.5445 - 6.36363 = 6.36363 - 3.1815 = 3.1815

3.1815
0.80886
0.477225
0.31815
0.159075


14.134725 + 6.36363 = 20.4977 (lower bound)

First estimate, using the zeta function directed to the left, the lower bound

20.4977 + 0.80886 = 21.30656 (upper bound)

2.7834 

22.29945 

Upper bound of first estimate using the zeta function directed to the right.

2.0757

20.22945

Lower bound 

Both lower and upper bounds of the estimates both zeta functions  will be used to refine the approximation.


Since the lower bound of the second zeta function has a smaller value than the corresponding figure of the lower bound of the first zeta function, the next value in the five element subdivision is substracted to get an UPPER BOUND.

10.9478
2.7834
1.6422
1.90478
0.5474

8.1644
2.0757
1.22466

0.81644
040822


1.22466

21.069425

This is the new UPPER BOUND for the approximation.

3.1815
0.80886
0.477225
0.31815
0.159075

20.4977 + 0.477225 = 20.975

2.0757 - 1.22466 = 0.85104

0.85104
0.21637
0.127656
0.085104
0.042552

Substracting these bottom four values successively from 21.069425:

20.853055
20.94177
20.984321
21.026873

Since now 20.984321 exceeds the estimate from the other zeta function (20.975), this will be new LOWER BOUND of the approximation.

So far:

Lower bound: 20.984321
Upper bound: 21.069425


0.80886 - 0.477225 = 0.331635

0.331635
0.084315
0.049745
0.0331635
0.016582

Adding 0.084315 to 20.975 will equal 21.0593, a figure which already exceeds the upper bound.

Adding 0.049745 to 20.975 will equal 21.024745.

Adding 0.0331635 to 20.975 will equal 21.0081635.

21.024745 will be the new upper bound of the approximation.

0.085104 - 0.042552 = 0.042552

0.042552
0.01081842
0.0063828
0.0042552
0.0021276

Substracting the bottom four figures from 21.026873 we obtain:

21.016055
21.0205
21.02262
21.024745

21.024745 is the SAME VALUE obtained from the five element subdivision for the first zeta function, this is how we know it is the upper bound of the entire approximation.

The lower bound is 21.016055.

To get the lower bound for the first zeta function, we have to subdivide the interval further.

The last estimate was 21.0081635.

0.084315 - 0.049745 = 0.03457

Using only the first two subdivision values:

0.03457
0.0087891

21.024545 + 0.0087891 = 21.03353, a figure which is too large.

0.049745 - 0.0331635 = 0.016582

Again, using only the first two subdivision values:

0.016582
0.0042157

21.0081635 + 0.0042157 = 21.01238

Continuing in this way we obtain:

21.01786

This will be the new lower bound of the entire approximation.

Continuing even further:

21.0226217 (this corresponds to the subdivision 0.00285253 and 0.00072523; the previous subdivision is 0.003825 and 0.00097247).

This is the same value as the one obtained from the other subdivision.

This will be new UPPER BOUND of the entire approximation.

0.0063828 - 0.0042552 = 0.0021273

0.0021273
0.000540845
0.0003181
0.00021273
0.000106365

21.02262 - 0.000540845 = 21.02207916

21.02262 - 0.00021273 = 21.02240727

To get the new lower bound, a figure higher than 21.016055 has to be obtained from the first zeta function subdivision.

0.003825
0.0009725
0.00057375
0.0003825
0.00019125

The value corresponding to 0.0009725 is 21.0218965.

This now is the new lower bound.


So far:

21.0218965 = lower bound

21.0226217 = upper bound

0.00285253 - 0.00072523 = 0.0021273

This is the same value as that obtained earlier from the second zeta function.

Since 21.02207916 exceeds 21.0218965, it will become the new UPPER BOUND of the entire approximation.

0.0009725 - 0.00057375 = 0.00039875

0.00039875
0.000101378

21.0218965 + 0.000101378 = 21.02199788

This figure will be the new lower bound.

The true value for the second zeta zero is:

21.022039639

Already we have obtained a five digit/three decimal place approximation:

21.02207916

 

z2 = 21.022

L(z2) = 5.2026

21.022 + 5.2026 = 26.2246

The third zeta zero, to four decimal places accuracy, using only the five elements subdivision applied to both zeta functions as a guide.

63.636363
16.1773
9.5445
6.36363
3.1815

14.134725 + 9.545 = 23.6747

16.1773 - 9.5445 = 6.6328

6.6328
1.68632
0.99492
0.66328
0.33164

23.6747 + 1.68632 = 25.36602

23.6747 + 0.99492 = 24.67462

23.6747 is the first lower bound.

25.36602 is the first upper bound.
 

3.7322

25.099425

1.64

23.459425

1.09478

24.001945

0.5474

24.552025

The values are taken from the subdivision:

14.68
3.7322
2.202
1.468
0.734

14.68 - 3.7322  = 10.9478

10.9478
2.7834
1.6422
1.09478
0.5474

Upper bound of first estimate using the zeta function directed to the right:

25.099425

Lower bound:

24.552025

Just like before, we search for a higher lower bound in both subdivisions, and for a lower upper bound in both subdivisions (a comparison, in order to locate the precise and correct subdivision interval for the zeta zero).

So far:

Upper bound

25.099425

Lower bound

24.67462


1.68632 - 099492 = 0.6914

0.6914
0.17578
0.10371
0.06914
0.03457

Adding these bottom four values successively to 24.67462:

24.8504
24.7783
24.77376
24.7092

0.6914 - 0.17578 = 0.51562

0.51562
0.1311
0.077343
0.051562
0.025781

Adding these bottom four values successively to 24.8504:

24.9815
24.9277
24.902
24.8762

From the other zeta function:

0.5474
0.139171
0.08211
0.05474
0.02737

Substracting these bottom four values successively from 25.099425:

24.96025
25.0173
25.044685
25.072055

24.96025 is the new LOWER BOUND.

Since 24.9815 (from the other zeta function) is a higher lower bound, this value will become the new lower bound for the entire approximation.

In order to obtain the new upper bound:

0.51562 - 0.1311 = 0.38452

0.38452
0.09776
0.057678
0.038452
0.019226

Adding these bottom three values successively to 24.9815:

25.039178
25.019952
25.000726

Then, 25.000726 becomes the new lower bound, while 25.019952 is the new upper bound for the first zeta function.

Since 25.0173 (second zeta function) is a lower value than 25.019952, this then is the new UPPER BOUND for the entire approximation.

So far:

25.000726 is the lower bound

25.0173 is the upper bound

0.038452 - 0.019226 = 0.019226

0.019226
0.004888
0.002884
0.0019226
0.0009613

Adding these bottom four values successively to 25.000726:

25.005614
25.00361
25.002649
25.00169

Using the second zeta function:

0.139171 - 0.08211 = 0.057061

0.057061
0.0145072
0.00856
0.0057061
0.00285305

Substracting these bottom four values successively from 25.0173:

25.0028
25.00874
25.0116
25.014447

The new lower bound is 25.00874 (a higher lower bound than 25.005614).

The new upper bound is 25.0116.

0.019226 - 0.004888 = 0.014338

0.014338 
0.0036453
0.0021507
0.0014338
0.0007169

Adding these bottom four values successively to 25.005614:

25.00926
25.00776
25.00705
25.006331

0.014338 - 0.0036453 = 0.0106927

0.0106927
0.0027185
0.001604
0.00106927
0.000534635

Adding these bottom four values to 25.00926:

25.012
25.010864
25.01033
25.0098

25.010864 is the new upper bound (a lower upper bound than 25.0116).

25.01033 is the new lower bound.

The true value for the third zeta zero is:

25.01085758

Already we have obtained a six digit/four decimal place approximation:

25.010864

 

z3 = 25.0108

L(z2) = 4.54832

29.55912

The fourth zeta zero, to three decimal places accuracy, using only the five elements subdivision applied to both zeta functions as a guide.

63.636363
16.1773
9.5445
6.36363
3.1815

14.134725 + 16.1773 = 30.312

16.1773 + 2.373 = 32.685

5.0045

28.81

2.95

30.8694

30.312 is the first lower bound.

30.8694 is the first upper bound.

2.373
0.60331
0.35595
0.2373
0.11865

Adding the bottom three values to 30.312:

30.668
30.55
30.43065

5.0045 - 2.95266 = 2.05184

2.05184
0.509798
0.307776
0.205184
0.102592

Substracting the bottom four values from 30.8694:

30.3596
30.5616
30.664216
30.76681

30.3596 is the new lower bound.

30.5616 is the new upper bound for the second zeta function.

30.43065 is the new upper bound for the first zeta function; since this figure is smaller than 30.5616, it is the upper bound of the entire approximation.

0.11865
0.0301656
0.0177975
0.011865
0.0059325

Adding the bottom four values to 30.312:

30.34216
30.3298
30.324
30.318

0.11865 - 0.0301656 = 0.0884844

Using only the first two subdivisions (corresponding to 534 and 136.1):

0.0884844
0.022496

30.342 + 0.022496 = 30.36465


The subdivisions for the second zeta function.

0.509796 - 0.307776 = 0.202022

0.202022
0.051362

30.5616 - 0.051362 = 30.510238

In order to make a new comparison between the two zeta functions, we have to subdivide further in order to determine the correct upper and lower bounds using subdivisions which have a very close value.

0.202022 - 0.051362 = 0.15066

0.15066
0.038304

30.510238 - 0.038304 = 30.47193

0.15066 - 0.038304 = 0.112356

0.112356
0.0285654

30.47193 - 0.0285654 = 30.44336

0.112356 - 0.0285654 = 0.083791

0.083791
0.021303
0.012569
0.0083791
0.0041895

Substracting the bottom four values from 30.44336:

30.422057
30.43079
30.435
30.43917


The subdivisions for the first zeta function.

0.0884844
0.022496

30.342 + 0.022496 = 30.36465

0.0884844 - 0.022496 = 0.065988

0.065988 
0.016777

30.36465 + 0.016777 = 30.38143

0.065988 - 0.016777 = 0.049211

0.049211
0.0125114

30.38143 + 0.0125114 = 30.39394

0.049211 - 0.0125114 = 0.0366996

0.0366996
0.00933051

30.39394 + 0.00933051 = 30.40327

0.0366996 - 0.00933051 = 0.0273691

0.0273691 
0.0069583

30.40327 + 0.0069583 = 30.410228

0.0273691 - 0.0069583 = 0.0204108

0.0204108
0.005189242

30.410228 + 0.005189242 = 30.41542

0.0204108 - 0.005189242 = 0.01522156

0.01522156 
0.00387

30.41542 + 0.00387 = 30.41928

0.01522156 - 0.00387 = 0.01135156

0.01135156
0.002886

30.41928 + 0.002886 = 30.422176

By comparison with the subdivisions obtained from the second zeta function, we can see that 30.422176 is the new lower bound.

0.01135156 - 0.002886 = 0.0084656

0.0084656
0.0021523

30.422176 + 0.0021523 = 30.424328


Returning to the subdivisions for the second zeta function.

0.112356 - 0.0285654 = 0.083791

0.083791
0.021303
0.012569
0.0083791
0.0041895

Substracting the bottom four values from 30.44336:

30.422057
30.43079
30.435
30.43917

0.021303 - 0.012569 = 0.008734 (this is the interval of the subidivision where the upper and lower bounds of the second zeta function are located)

0.008734
0.0022205
0.0013101
0.0008734
0.0004367

Substracting the bottom values from 30.43079:

30.42857
30.42948
30.429917
30.43035


Returning to the subdivisions for the first zeta function.

0.0084656
0.0021523

30.422176 + 0.0021523 = 30.424328

0.0084656 - 0.0021523 = 0.0063133

0.0063133
0.0016051
0.000947
0.00063133
0.000315665

Adding the bottom three values to 30.424328:

30.425275
30.424959
30.424684


Returning to the subdivisions for the second zeta function.

0.008734 - 0.0022205 = 0.0065135

0.0065135
0.001656

30.42857 - 0.001656 = 30.4269

0.0065135 - 0.001656 = 0.0048575

0.0048575
0.001235

30.4269 - 0.001235 = 30.42566

0.0048575 - 0.001235 = 0.0036255

0.0036255
0.000921
0.00054383
0.00036255
0.0001813

Substracting the four bottom values from 30.42566:

30.42474
30.42512
30.4253
30.42548


30.424684 is the new lower bound.

30.424959 is the new upper bound.

The true value for the fourth zeta zero is:

30.424876126

Already we have obtained a five digit/three decimal place approximation:

30.424684

(to be continued)

 

 

Let us remember that I am only using basic arithmetic to derive the values of the zeta zeros, in accordance with the optimum interval and subsequent subdivision, and using BOTH zeta functions on that same interval to obtain new upper and lower bounds for the zeros.

 

z4 = 30.4247

L(z4) = 3.98331

34.408

The fifth zeta zero, to three decimal places accuracy, using only the five elements subdivision applied to both zeta functions as a guide.

63.636363
16.1773
9.5445
6.36363
3.1815
 

16.1773 + 2.373 = 32.685

4.7459 - 2.373 = 2.373

2.373
0.6033
0.356
0.2373
0.118645

Adding to the bottom four values to 32.685:

33.2883
33.041
32.9223
32.8036


1.968

31.8494

0.984

32.8294

6.7106

33.8

32.8294 is the first lower bound.

Since 32.9223 is a higher lower bound, this value is the lower bound of the entire approximation.

To find the first upper bound, we need to subdivide the intervals for the second zeta function further, in order to find a lower upper bound than 33.041.

0.98422
0.25023

33.8 - 0.25023 = 33.55

0.98422 - 0.25023 = 0.734

0.734
0.18661

33.55 - 0.18661 = 33.364

0.734 - 0.18661 = 0.5474

0.5474
0.139171

33.364 - 0.139171 = 33.225

0.5474 - 0.139171 = 0.40823

0.40823
0.103788

33.225 - 0.103788 = 33.1212

0.40823 - 0.103788 = 0.304442

0.304442
0.0774

33.1212 - 0.0774 = 33.0438

0.304442 - 0.0774 = 0.227042

0.227042
0.05772

33.0438 - 0.05772 = 32.9861

32.9861 is the new upper bound of the entire approximation.


0.356 - 0.23729 = 0.11871

0.11871
0.0302
0.01781
0.011871
0.0059355

Adding the bottom four values to 32.9223:

32.9525
32.9401
32.9342
32.928

32.9401 is the new upper bound.


Returning to the subdivisions for the second zeta function.

0.227042 - 0.05772 = 0.16932

0.16932
0.04305

32.9861 - 0.04305 = 32.94305

0.16932 - 0.04305 = 0.12627

0.12627
0.0321
0.01894
0.012627
0.0063135

Substracting the bottom four values from 32.94305:

32.911
32.9241
32.9304
32.93673

32.93672 is the new upper bound.

0.012627 - 0.0063135 = 0.0063135

0.0063135
0.0016052
0.000947
0.00063135
0.000315675

Substracting the bottom four values from 32.93673:

32.935125
32.935783
32.9361
32.936414


Returning to the subdivisions for the first zeta function.

0.01781 - 0.011871 = 0.0059355

0.0059355
0.001509
0.000891
0.00059355
0.000297

Adding the bottom four values to 32.9342:

32.93571
32.935091
32.9348
32.9345

Since 32.935091 is a lower value than 32.935125, this figure is the new upper bound of the entire approximation.

0.0063135 - 0.0016052 = 0.0047083

0.0047083
0.00119704
0.000706245
0.00047083
0.000235415

Substracting the last figure from 32.935125 we obtain 32.93489.

Since this is greater value than 32.9348, it becomes the new lower bound of the entire approximation.

This is further proof that 32.935125 was an upper bound, and that 32.935091 is the new upper bound for the entire approximation.

The true value for the fifth zeta zero is:

32.935061588

Already we have obtained a five digit/three decimal place approximation:

32.935091


Further subdivisions for greater accuracy.

0.00047083 - 0.000235415 = 0.000235415

0.000235415 
0.000059852
0.0000353
0.0000235415
0.000011771

Substracting the bottom four values from 32.935125:

32.935065
32.935089
32.935101
32.935113


Returning to the subdivisions for the first zeta function.

0.000891 - 0.00029745 = 0.00029745

0.00029745
0.000075624

32.9348 + 0.000075624 = 32.9348756

0.00029745 - 0.000075624 = 0.000221826

0.000221826
0.0000564

32.9348756 + 0.0000564 = 32.93492

0.000165426
0.000042055

32.93492 + 0.000042055 = 32.934962

0.00012337
0.000031366

32.934962 + 0.000031366 = 32.9349934

0.000092334
0.000023475

32.9349934 + 0.000023475 = 32.93501688

0.000068859
0.0000175067

32.93501688 + 0.0000175067 = 32.9350344

0.000051353
0.000013056

32.9350344 + 0.000013056 = 32.93504746

0.000038297
0.00000973663

32.93504746 + 0.00000973663 = 32.9350572

0.000028561
0.00000726135

32.9350572 + 0.00000726135 = 32.93506446

This becomes the new upper bound of the entire approximation (a value smaller than 32.935065 obtained from the second zeta function subdivision).

0.000028561
0.00000726135
0.00000428415

32.9350572 + 0.00000428415 = 32.93506148

The true value for the fifth zeta zero is:

32.935061588

Already we have obtained an eight digit/six decimal place accuracy:

32.93506148
 

z5 = 32.935

L(z5) = 3.7927

36.7277

The sixth zeta zero, to three decimal places accuracy, using only the five elements subdivision applied to both zeta functions as a guide.

63.636363
16.1773
9.5445
6.36363
3.1815
 

14.134725 + 16.1773 + 7.1185 = 37.43

12.066 - 7.1185 = 4.9475

4.9475
1.2577
0.74205
0.49475
0.24735

37.43 + 0.24735 = 37.67735

2.64

37.8794

3.96

36.56942

37.6773 is the first upper bound.

36.56942 is the first lower bound.


0.24735
0.062886
0.0371
0.024725
0.012367

Adding the bottom four values to 37.43:

37.4929
37.467
37.455
37.442

0.24735 - 0.062886 = 0.184464

0.184464
0.0469

37.4929 + 0.0469 = 37.54

0.184464 - 0.0469 = 0.137564

0.137564
0.034974
0.02063
0.0137564
0.00688

Adding the bottom four values to 37.54:

37.575
37.5606
37.553756
37.54688

37.575 is the new lower bound.


3.96 - 2.6395 = 1.3205

1.3205
0.335724
0.198075
0.13205
0.066025

Substracting the bottom values from 37.8794 (the upper bound for the second zeta function):

37.54367
37.681
37.74735
37.81337

0.335724 - 0.198075 = 0.137649

0.137649
0.03499581
0.02064735
0.0137649
0.00688245

Substracting the bottom four values from 37.681:

37.6460042
37.66035
37.66724
37.6741

0.137649 - 0.03499581 = 0.1026532

0.1026532
0.0261

37.6460042 - 0.0261 = 37.61991

0.1026532 - 0.0261 = 0.0765532

0.0765532 
0.019463

37.61991 - 0.019463 = 37.600447

0.0765532 - 0.019643 = 0.0571

0.0571
0.0145146
0.008565
0.0057
0.002855

Substracting the bottom four values from 37.600447:

37.5859324
37.592
37.5947
37.5976

37.5859324 is the new lower bound of the entire approximation.


Returning to the subdivisions for the first zeta function.

0.137564 - 0.034974 = 0.10259

0.10259
0.0260825
0.0153885
0.010259
0.0051285

Adding the bottom four values to 37.575:

37.601
37.5904
37.58526
37.58013

37.5904 is the new upper bound.

The true value for the sixth zeta zero is:

37.586178159

Already we have obtained a five digit/three decimal place approximation:

37.5859324

The upper bound can also be used within the same derivation to obtain new estimates.

14.134725 + 16.1773 + 7.1185 = 37.43

12.066 - 7.1185 = 4.9475

4.9475
1.2577
0.74205
0.49475
0.24735

37.43 + 0.24735 = 37.67735

0.24735
0.062886
0.0371
0.024725
0.012367

Adding the bottom four values to 37.43:

37.4929
37.467
37.455
37.442


Now, the same values will be substracted from the upper bound, 37.6773:

0.062886
0.0371
0.024735
0.012367

Obtaining:

37.6144
37.6402
37.6525
37.665


The second lower bound for the second zeta function is 37.54367.

Now, this value will be used to add the corresponding values belonging to the derivation of the first zeta function.

0.062886
0.0371
0.024735
0.012367

Adding these values (belonging to the first zeta function subdivisions) to 37.54367:

37.6066
37.581
37.568
37.556


Conversely, the upper bound of the first zeta function, 37.6773, will be used to get new estimates belonging to the second zeta function.

1.3205
0.335724
0.198075
0.13205
0.066025

Substracting the bottom four values from 37.6773:

37.3416
37.479
37.545
37.6113


The lower bound can also be used within the same derivation to obtain new estimates.

0.137649
0.03499581
0.02064735
0.0137649
0.00688245

Adding the bottom four values to 37.54367:

37.5786
37.56432
37.5574
37.5505


The new estimates are: 37.545 as a lower bound, 37.6113 as an upper bound.

Then, 37.6066 becomes the new upper bound, while 37.581 is the new lower bound.

By observation, 37.600447 (the value previously calculated) becomes the new upper bound of the entire approximation.

Then, 37.5904, the value from the first zeta function subdivision is the new upper bound, while 37.58526 is the new lower bound.


Thus, these new features/results greatly simplify the entire sequence of five elements subdivisions estimates: now one can also add/substract the upper/lower bounds as needed, and use an estimate from the first zeta function (or from the second zeta function) as an upper/lower bound starting point value to use in the subdivisions calculations for the second zeta function (or for the first zeta function).

z6 = 37.586

L(z6) = 3.5126

41.098

The seventh zeta zero, to three significant digits accuracy, using only the five elements subdivision applied to both zeta functions as a guide.

63.636363
16.1773
9.5445
6.36363
3.1815

 

14.134725 + 16.1773 + 7.1185 = 37.43

12.066 - 7.1185 = 4.9475

4.9475
1.2577
0.74205
0.49475
0.24735

37.43 + 1.2577 = 38.6877

4.9475 - 1.2577 = 3.6898

3.6898
0.9381
0.55347
0.36898
0.1845

Adding the bottom four values to 38.6877:

39.6258
39.24117
39.05668
38.8722


5.309

44.2 

8.998

40.52

40.52 is the first lower bound.

44.2 is the first upper bound (even though 44.2 is greater than the eighth zeta zero, 43.327)


Now, the new features/results from the previous message on this page will be used.

8.998 - 5.309 = 3.689

3.689
0.9379
0.55335
0.3689
0.18445

Adding the bottom four values to 40.52:

41.458
41.073
40.889
40.704

40.889 is the new lower bound.

41.073 is the new upper bound.

38.6877 + 3.6898 = 42.378

1.2577
0.74205
0.4947
0.24735

Substracting 1.2577 from 42.378 we obtain 41.1203.

3.6898
0.9381
0.55347
0.36898
0.1845

Substracting the bottom four values from 41.1203:

40.1823
40.567
40.7513
40.9358

Since 40.9358 is a smaller value than 41.073, 40.9358 is the new upper bound of the entire approximation.

The true value for the seventh zeta zero is: 40.918719012

Already we have obtained a three significant digit approximation:

40.9358


A more difficult approach, without using the new features/results, would be use the five element subdivision algorithm, starting with 44.2 for the second zeta function (44.2, 43.262, 42.5632, 42.04172, 41.6526, 41.3624, 40.5112 and 41.146, 40.9846, 40.9136 and 40.93726) and continuing with the value of 38.6877 for the first zeta function (39.6258, 40.3254, 40.847126 and 40.9236, 40.86658, 40.8811, 40.892, 40.90007).

Once the four subdivision figures are obtained, there is no need to even bother to find the value of the corresponding zeta zero: all that matters are the five element subdivision points, then the zeta zero can be computed effortlessly if so desired.

 

z7 = 40.9187

L(z7) = 3.353

44.272

The eighth zeta zero, to three significant digits accuracy, using only the five elements subdivision applied to both zeta functions as a guide.

63.636363
16.1773
9.5445
6.36363
3.1815

 

14.134725 + 16.1773 + 12.066 = 42.378

14.134725 + 16.1773 + 12.066 + 1.77 = 44.148


5.309

44.2

8.998

40.51


1.77
0.45
0.2655
0.177
0.0885

Adding the bottom four values to 42.378 (which is the first lower bound):

42.83
42.64
42.555
42.46


Substracting the bottom four values from 44.148 (the first upper bound), thus using the new results/features posted on this page:

43.7
43.88
43.97
44.06


For the second zeta function:

8.998 - 5.309 = 3.689

3.689
0.9379
0.55335
0.3689
0.18445

Substracting the bottom four values from 44.2:

43.262
43.646
43.8311
44.015

Adding the bottom four values to 42.378:

43.316
42.93
42.747
42.562

42.83 is the new lower bound; since 43.316 has a greater value than 42.83, 43.316 is the new lower bound for the entire approximation.

43.7 is the new upper bound.

At this point, a three digit approximation has already been obtained (true value of the eighth zeta zero is 43.327073281); however, the five element subdivision algorithm will be continued, in order to show the precise calculations.


Since 43.262 is the lower bound for the second zeta function (while 43.646 is the upper bound), we already know that the true value of the eighth zeta zero is to be found in the 0.9379 - 0.55335 interval.


3.689 - 0.9379 = 2.7511

2.7511
0.699
0.4126
0.27511
0.1375

Substracting the bottom four values from 43.262:

42.563
42.85
42.987
43.124

Adding the bottom four values to 43.316:

44.015
43.73
43.6
43.45


0.9379 - 0.55335 = 0.38455

0.38455
0.0977
0.0577
0.038455
0.01923

43.646 - 0.0977 = 43.55

0.38455 - 0.0977 = 0.28685

0.28685
0.07293

43.55 - 0.07293 = 43.477

0.28685 - 0.07293 = 0.21392

0.21392
0.0544

43.477 - 0.0544 = 43.42

0.21392 - 0.0544 = 0.15952

0.15952
0.0405

43.42 - 0.0405 = 43.38

0.15952 - 0.0405 = 0.11902

0.11902
0.03026

43.38 - 0.03026 = 43.35

0.11902 - 0.03026 = 0.08876

0.08876
0.022566

43.35 - 0.022566 = 43.327 (a five digit approximation)


Returning to the calculations for the first zeta function.

1.77 - 0.45 = 1.32

1.32
0.3356
0.198
0.132
0.066

Adding the bottom four values to 42.83:

43.1656
43.028
42.962
42.896

Substracting the bottom four values from 43.7:

43.364
43.502
43.568
43.634

1.32 - 0.3356 = 0.9844

0.9844
0.2503
0.14706
0.09844
0.04922

Adding the bottom four values to 43.1656:

43.416
43.315
43.264
43.215

Substracting the bottom four values from 43.364:

43.114
43.216
43.26
43.315

0.9844 - 0.2503 = 0.7341

0.7341
0.1866
0.11
0.07341
0.0367

Adding the bottom four values to 43.416:

43.6
43.526
43.49
43.45


Returning to the calculations for the second zeta function:

0.1375
0.034958
0.0206
0.01375
0.006875

Adding the bottom four values to 43.316:

43.351
43.2826
43.276
43.268


Successively, the new upper bounds are: 43.634, 43.6, 43.568, 43.526, 43.502, 43.45, 43.416, 43.364.

The new lower bounds are: 42.83, 42.93, 42.987, 43.028, 43.215, 43.26, 43.316.

The true value for the eighth zeta zero is: 43.327073281

Already we have obtained a three significant digit approximation:

43.316
 

The fact that the five element subdivisions algorithm can be applied to each separate 63.6363... segment can immediately be used to great advantage to calculate the zeta zeros  for extremely large values of t (1/2 + it). So far, the computations of the Riemann zeta function for very high zeros have progressed to a dataset of 50000 zeros in over 200 small intervals going up to the 1036-th zero.

The main problem is the calculations of the exponential sums in the Riemann-Siegel formula.

However, the five element subdivisions algorithm suffers from no such restrictions.

The 63.6363... segment can be shifted to any desired height, using arbitrary-precision arithmetic.

Therefore, computations of zeros around the first Skewes number, 1.39822 x 10316 become possible using the Schönhage–Strassen algorithm for the multiplication/addition of very large numbers.

The Riemann-Siegel requires the addition of all of the terms in the formula, involving the evaluation of cosines, logarithms, square roots, and a complex set of remainders.

With the five element subdivision algorithm, only the following calculations are required: k x 63.6363..., where k can be 1.39822 x 10316 or 1010,000 (10 followed by ten thousand zeros). No divisions are required, no evaluation of elementary transcendental or algebraic functions is needed. The five element sequence of proportions are T, 63.6363... x T/250, 3T/10, T/10, T/20: simple multiplications by 1/250, 3/10, 1/10 and 1/20.

The only figure remaining to be calculated very precisely is the actual value of the interval distance.

14/22 = 0.63636363...

2/π = 0.636619722...

286.1/450 = 0.6357777...

14.134725 x 45 = 636.062625....

π has been calculated to over one million digits, the first zeta zero to over 40,000 digits.

The precise figure can be deduced by using the five element subdivision algorithm to the following heights: 636.63, 6,363.63, 63,636.63, 636,363.63.


Two examples which prove that the 63.6363 segment can be shifted to higher intervals on the critical 1/2 line, with no previous knowledge of the values of the other zeta zeros.

Zeta zero: 79.337375020

14.134725 + 63.63 = 77.7647

L(77.7647) = 2.4975 (average spacing estimate 80.262)

77.7647 + 0.80886 = 78.57356

77.7647 + 3.1815 = 80.9462

141.3947 - 16.1773 - 12.066 - 8.998 - 6.7106 - 5.0045 - 3.7322 - 2.7834 - 2.0757 - 1.548 - 1.1544 - 0.861 = 80.2836

141.3947 - 16.1773 - 12.066 - 8.998 - 6.7106 - 5.0045 - 3.7322 - 2.7834 - 2.0757 - 1.548 - 1.1544 - 0.861 - 0.692 = 79.598

141.3947 - 16.1773 - 12.066 - 8.998 - 6.7106 - 5.0045 - 3.7322 - 2.7834 - 2.0757 - 1.548 - 1.1544 - 0.861 - 0.692 - 0.4661 = 79.1318

3.1815 - 0.80886 = 2.3726

2.3726
0.60332
0.3559
0.23726
0.11863

Adding the bottom four values to 78.57356:

79.177
78.929
78.811
78.6922

79.177 is the first lower bound.

2.3726 - 0.60322 = 1.7694

1.7694
0.45
0.26541
0.17694
0.08847

Adding the bottom four values to 79.177:

79.627
79.442
79.354
79.2655

79.598 is the first upper bound.


0.4661 
0.275
0.1833
0.09

Substracting these values from 79.598:

79.1318
79.323
79.4147
79.508

79.323 is the new lower bound. 

79.354 is the new upper bound.

0.17694 - 0.08847 = 0.08847

0.08847 
0.0225

79.2655 + 0.0225 = 79.288

0.08847 - 0.225 = 0.06597

0.06597
0.016772

79.288 + 0.016772 = 79.30437

0.06597 - 0.016772 = 0.0492

0.0492
0.01251

79.30437 + 0.01251 = 79.3173

0.0492 - 0.01251 = 0.03669

0.03669
0.009328

79.3173 + 0.009328 = 79.32663

79.362663 is the new lower bound.

0.03669 - 0.009328 = 0.027362

0.027362
0.0069565
0.0041043
0.0027362
0.00131681

Adding the bottom four values to 79.3266:

79.3336
79.3307
79.32937
79.328

0.027362 - 0.0069565 = 0.0204055

0.0204055
0.0051879
0.003061
0.00204055
0.00102

Adding the bottom four values to 79.3336:

79.33879
79.336661
79.33564
79.33462

0.0204055 - 0.0051879 = 0.0152176

0.0152176
0.003869
0.00283
0.00152176
0.000761

Adding the bottom four values to 79.33879:

79.34266
79.3416
79.3403
79.3395


The calculations for the second zeta function.

0.275 - 0.1833 = 0.0917

0.0917
0.0233

79.4147 - 0.0233 = 79.3914

0.0917 - 0.0233 = 0.0684

0.0684
0.0174

79.3914 - 0.0174 = 79.374

0.0684 - 0.0174 = 0.051

0.051
0.012966

79.374 - 0.012966 = 79.361

0.051 - 0.012966 = 0.038034

0.038034
0.00967

79.361 - 0.00967 = 79.35133

79.35133 is the new upper bound.

0.038034 - 0.00967 = 0.028364

0.028364 
0.00721

79.35133 - 0.00721 = 79.34412

0.028364 - 0.00721 = 0.021154

0.021154
0.0053782
0.003773
0.0021154
0.001058

Substracting the bottom values from 79.34412:

79.338742
79.34095
79.342
79.34306

0.021154 - 0.0053782 = 0.015776

0.015776
0.004011
0.0023664
0.0015776
0.000789

Substracting the bottom four values from 79.338742:

79.33473
79.3364
79.337164
79.33795

Now, the new features/results from the previous message will be used.

0.0917
0.0233
0.013755
0.00917
0.004585

Adding the bottom four values to 79.323:

79.3276
79.3322
79.3367
79.3453

0.17694 - 0.08847 = 0.08847 

0.08847
0.022493
0.013271
0.008847
0.0044235

Substracting the bottom four values from 79.354:

79.33151
79.34073
79.3455
79.3496

79.3367 is the new lower bound.

79.33879 is the new upper bound.

Since 79.337164 is a higher figure than 79.3367, 79.337164 is the new lower bound for the entire approximation.

Without any knowledge of the values of the previous zeta zeros, a five digit/three decimal place approximation of the zeta zeros was obtained.


Zeta zero: 143.111845808

14.134725 + 63.63 + 63.63 = 141.3947

L(141.3947) = 2.018 (average spacing estimate 143.4126)

141.3947 + 0.80886 = 142.20356

141.3947 + 0.60322 = 142.8068

141.3947 + 0.45 = 143.2568

141.3947 + 0.335 = 143.592

205.0247 - 16.1773 - 12.066 - 8.998 - 6.7106 - 5.0045 - 3.7322 - 2.7834 - 2.0757 - 1.548 - 1.1544 - 0.861 = 143.9187

205.0247 - 16.1773 - 12.066 - 8.998 - 6.7106 - 5.0045 - 3.7322 - 2.7834 - 2.0757 - 1.548 - 1.1544 - 0.861 - 0.692 = 143.2277

205.0247 - 16.1773 - 12.066 - 8.998 - 6.7106 - 5.0045 - 3.7322 - 2.7834 - 2.0757 - 1.548 - 1.1544 - 0.861 - 0.692 - 0.4661 = 142.7618

142.8068 is the first lower bound.

143.2277 is the first upper bound.

1.7694
0.45
0.26541
0.17694
0.08847

Adding the bottom four values to 142.8068:

143.2568
143.07221
142.984
142.8953


0.4661 
0.275
0.1833
0.09

Substracting these values from 143.2277:

142.7618
142.9527
143.0444
143.1377

Now, the new features/results from the previous message will be used.

1.7694
0.45
0.26541
0.17694
0.08847

Substracting the bottom three values from 143.2568:

142.9914
143.0798
143.16833


0.4661 
0.275
0.1833
0.09

Adding the bottom three values to 142.7618:

143.0368
142.945
142.8518

143.07221 is the new lower bound.

143.1377 is the new upper bound.

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Global formulas for Lehmer pairs and Large Gaps

 

Zeta zeros distribution: regular zeros, large gaps between zeros, Lehmer pairs and strong Lehmer pairs.

The most difficult aspect of the distribution of the zeta zeros is the location of the strong Lehmer pairs.

leh02.jpg

 

There can’t be a zero of ζ'(s) between every pair of zeros of ζ(s) because the density of zeros of ζ(s) is log(T/2π)/2π while the density of zeros of ζ'(s) is log(T/4π)/2π. So on average there is a “missing” zero of ζ'(s) in each T interval of width 2π/log 2 ≈ 9.06.

ROOTS OF THE DERIVATIVE OF THE RIEMANN ZETA FUNCTION

https://arxiv.org/pdf/1002.0372.pdf

On Small Distances Between Ordinates of Zeros of ζ(s) and ζ'(s)

http://math.boun.edu.tr/instructors/yildirim/paper/OnSmallDistancesBtwOrdinates.pdf

LEHMER PAIRS AND DERIVATIVES OF HARDY’S Z-FUNCTION

https://arxiv.org/pdf/1612.08627.pdf

The author has calculated that the first two million zeros include 4637 pairs of zeros which satisfy the first assertion, while 1901 pairs actually belong to the set L.


LEHMER PAIRS REVISITED

https://arxiv.org/pdf/1508.05870.pdf

In other words, strong Lehmer pairs tend to arise from a small gap between zeros of ζ(s), and from the zeros of ζ'(s) very near the critical line.

leh01.jpg

Figure 2 shows the argument of ζ'(s)/ζ(s), interpreted as a color, in a region which includes Lehmer’s example. The Riemann zeros 1/2 + iγ6709 and 1/2 + iγ6710 are now poles, while in between we see a zero of ζ'(s) at 0.50062354 + 7005.08185555i, very close to the critical line, even on the scale of this close pair of Riemann zeros.

 

Global formula for Lehmer pairs/close values of the pairs of zeta zeros


T =~ {n ⋅ 2π/ln2 + n ⋅ 2π/ln2 + π/ln2}/2

T =~ {n ⋅ 2π/ln2 + π/ln2 + (n + 1) ⋅ 2π/ln2 }/2


n > 2

T will always be part of an infinite sequence of Lehmer pairs (which includes also strong Lehmer pairs).

 

Large gaps formula for the zeta function zeros


32 + 25 x n

16 + 25 x n

8 + 24 x n



There will always be large gaps right next to these values of the critical line.

 

14.134725142
     21.022039639
     25.010857580
     30.424876126
     32.935061588
     37.586178159
     40.918719012
     43.327073281
     48.005150881
     49.773832478
     52.970321478
     56.446247697
     59.347044003
     60.831778525
     65.112544048
     67.079810529
     69.546401711
     72.067157674
     75.704690699
     77.144840069
     79.337375020
     82.910380854
     84.735492981
     87.425274613
     88.809111208
     92.491899271
     94.651344041
     95.870634228
     98.831194218
    101.317851006
    103.725538040
    105.446623052
    107.168611184
    111.029535543
    111.874659177
    114.320220915

Large gaps at:

16
24
32
40
48
56
64
72
80
88
96
104
112
120
128

    399.985119876
    401.839228601
    402.861917764
    404.236441800
    405.134387460
    407.581460387
    408.947245502
    410.513869193
    411.972267804
    413.262736070
    415.018809755
    415.455214996
    418.387705790

Large gaps at 400, 408 and 416.

 

Examples:

415.018809755
415.455214996

8 unit interval: 408 to 416

2π/ln2 + π/ln2 interval

403.38 to 412.445

π/ln2 interval

407.912 to 416.977

(416.977 + 412.445)/2 = 414.711


7005.062866175
7005.100564674

8 unit interval: 7000 to 7008

2π/ln2 interval

6997.964 to 7007.029

2π/ln2 + π/ln2 interval

7002.496 to 7011.56

(7007.029 + 7002.496)/2 = 7004.763


17143.786536184
17143.821843505

8 unit interval

17136 to 17144

2π/ln2 interval

17141.386 to 17150.451

2π/ln2 + π/ln2 interval

17145.918 to 17154.98

(17145.918 + 17141.386)/2 = 17143.65


35839.415210178
35839.746238617

8 unit interval

35832 to 35840

2π/ln2 interval

35832.8 to 35841.9

2π/ln2 + π/ln2 interval

35837.36 to 35846.42

(35837.36 + 35841.9)/2 = 35839.6


How to generate the 2π/ln2 intervals

9.06472
18.1294
27.194
36.258
45.323...

(we simply multiply 2π/ln2 by n, n = 1,2,3...)

How to generate the 2π/ln2 + π/ln2 intervals

(9.06472 + 18.1294)/2 =  13.5971

13.5971 - 2π/ln2 = 4.532360142

4.53236
13.5971
22.6618
31.726
41.4197...

(we simply shift the 2π/ln2 intervals by a factor of π/ln2)

 


2π/ln2 ⋅ 144(n + ε)

144 ⋅ ε = k, where k = 1,2,3...,143

2π/ln2 ⋅ (144n + k)



The previous formulas featured 2π/ln2 multiplied by n; this global formula incorporates the decimal parts as well, which have special values.

Examples:

17143.7865

17143.7865/2π/ln2 = 1891.2648

1891.2648/144 = 13.1338

144 x 0.1338 =~ 19

2π/ln2 x (144 x 13 + 19) = 17141.385

The equations derived previously are special cases of this global formula.


169872.853

2π/ln2 x (144 x 130 + 20) = 169872.858


45505.59

2π/ln2 x (144 x 34 + 124) = 45504.8944


45436.65

2π/ln2 x (144 x 34 + 117) = 45441.44


412597.295

2π/ln2 x (144 x 316 + 13) = 412598.86


555136.9163

2π/ln2 x (144 x 425 + 41) = 555132.52
2π/ln2 x (144 x 425 + 42) = 555141.58

Average = 555137.0511


7954022502373.43289015387

2π/ln2 x (144 x 6093543501 + 75) = 7954022502369.544
2π/ln2 x (144 x 6093543501 + 76) = 7954022502378.53

Average = 7954022502374.037


2414113624163.41943

1303.jpg

 

2π/ln2 x (144 x 1849442389 + 136) = 2414113624163.446


13066434408794226520207.1895041619

 

1301.jpg

1302.jpg

Since now T is very large (the average spacing is 0.128), the decimal parts of k also can be used (k = v + 1/2, v+ 1/4, v+ 3/4); in this case 144 x 0.1441 = 20.75).

With 20.75, we get:

2π/ln2 x (144 x 10010140964026289815 + 20.75) = 13,066,434,408,794,226,520,207.14279

131.jpg

 

8847150598019.22359827

1305.jpg

2π/ln2 x (144 x 6777765214 + 101) = 8847150598015.23188
2π/ln2 x (144 x 6777765214 + 102 = 8847150598024.2966

Average = 8847150598019.764243


2π/ln2 x 144 = 415.496 x π


7005.1

2π/ln2 x (144 x 5 + 53) = 7007.028

 

A shortcut formula for Lehmer pairs

(636.3 x 3n - 16.9 x n)2π/ln2

(n = 1,2,3...)

 

n = 1

17150.45078


http://www.dtc.umn.edu/~odlyzko/zeta_tables/zeros1

17143.786536184
17143.821843505

n = 2

34300.90155

34295.104944255
34295.371984027

n = 3

51451.35233

51448.076349964
51448.729475327
51449.153911623

n = 4

68601.80311

68597.636479797
68597.971396943

n = 5

85752.25388

85748.621773488
85748.861163006

n = 6

102902.7047

102907.166732245
102907.475751344

n = 7

120053.1554

120055.446373211
120055.565321075

 

2π/ln2 is the most important constant of the eta zeta function (alternating series zeta function):

https://arxiv.org/pdf/math/0209393.pdf

https://arxiv.org/pdf/0706.2840.pdf

This would be the starting point to prove the shortcut formula which really does provide the exact results.

 

Now, would using this shortcut formula for Lehmer pairs together with the global algorithm for finding zeta zeros be enough to prove the RH?

Not yet, since we need to prove that the shortcut formula will ALWAYS include a strong Lehmer pair in the sequence of Lehmer pairs.

However, we can accomplish something else.

The existence of infinitely many Lehmer pairs implies that the de Bruijn-Newman constant Λ is equal to 0.

Therefore, a constructive/computer-assisted proof of the Riemann hypothesis would be possible, if further Lehmer pairs can be produced with little computational effort.

I believe that the strong Lehmer pairs have shortcut formulas/special infinite sequences from which they can be generated with little effort.

The (636.3 x 3n - 16.9 x n)2π/ln2 infinite sequence certainly suggests that other similar sequences do exist.

Since now we no longer have to rely on the Riemann-Siegel formula to produce the zeta zeros, the calculation of zeros around the 1050, 10300, 101000 values on the 1/2 critical line become possible using the four subdivisions algorithm, the França-LeClair equation (ϑ(tn) + limδ→0+ arg ζ(1/2 + δ + itn) = (n - 3/2)π), used in conjunction with Backlund's method and Gram points.

That is, further Lehmer pairs can be produced with very little effort, using the two infinite sequences above: further sequences exist, which can capture the strong Lehmer pairs even better.

These Lehmer pairs then can be used to produce lower and lower bounds for the de Bruijn-Newman constant, finally proving that Λ is equal to zero.
 

 

The values of the strong Lehmer pairs behave like regular zeta zeros in a way: they exhibit large gaps and double Lehmer pairs (two pairs which are located very close to each other).

To understand the behavior of the regular zeta zeros, a certain interval was used (63.636363...), and we used the five elements subdivision algorithm to capture perfectly the value of each zeta zero.

The Lehmer pairs (see the definition used earlier) occur each and every 2π/ln2 units or at an average of the n x 2π/ln2 and (n + 1) x 2π/ln2 values.

Even this information can be used with great advantage, together with the five elements subdivision algorithm or with the França-LeClair equation to find the values of each and every Lehmer pair at very high figures on the 1/2 critical line, a feat which could not be accomplished before.

Strong Lehmer pairs tend to arise from a small gap between zeros of ζ(s), and from the zeros of ζ'(s) very near the critical line.


Interval for the strong Lehmer pairs

2π/ln2 x 100 sacred cubits

That is, we treat each 2π/ln2 value as a single unit of measure (a distance of 9.064720284... = one unit).

2π/ln2 x 100 sacred cubits = 576.84583...

Then, we subidivide this interval just like before using the 26.7, 53.4, 80, 136.1, 534 subdivisions, looking for the location of the strong Lehmer pairs.

576.84583
146.657
86.52676
57.6845
28.842

576.84583 - 146.657 = 430.88

430.88
109.371
64.5282

430.88 - 109.371 = 320.817

320.817
81.5645
48.1225

320.817 - 81.5645 = 239.2525

239.2525
60.827

239.2525 - 60.827 = 178.4255

178.4255
45.388
26.71
17.84
8.92

178.4255 - 45.388 = 133.0375

133.0375
33.823
19.955
13.303
6.65

133.0375 - 33.823 = 99.2145

99.2145
25.224

With these values, we obtain very nice approximations for the Lehmer pairs located at 111.03 and 415.45: 416.26 (first zeta function) and 419.37 (second zeta function) and 113.08 (first zeta function).

To capture the values of the higher strong Lehmer pairs, 7005.1 and 17143.78, the interval will be increased to 57684.52413 (2π/ln2 x 10000 sacred cubits).

Using the same subdivision, we get 7048.6 and 16950.5, as first values of entire sequence of approximations.

For the strong Lehmer pairs which have 12 digits, the interval becomes:

57684583623255.194152266 (2π/ln2 x 1 x 1012 sacred cubits)

or an even better approximation,

57684583623255.1941522669588143649472078575

This would be the only way to get approximate values of very large strong Lehmer pairs, and to gain an understanding of their location, which is not random, but has a very precise pattern, based on the 2π/ln2 x 100 sacred cubits interval.

 

 

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I find it unlikely that a solution to RH could be found without the use of mathematical analysis; that is, without the use of limiting arguments. RH involves quantifying over real and complex numbers, including noncomputable ones. That's my sense of the matter, not being a specialist in RH.

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Posted (edited)

It is my belief that RH is a genuinely arithmetic question that likely will not succumb to methods of analysis. Number theorists are on the right track to an eventual proof of RH, but we are still lacking many of the tools. 

J. Brian Conrey

 

It is clearly a preliminary note and might not have been written if L. Kronecker had not urged him to write up something about this work (letter to Weierstrass, Oct. 26 1859). It is clear that there are holes that need to be filled in, but also clear that he had a lot more material than what is in the note. What also seems clear : Riemann is not interested in an asymptotic formula, not in the prime number theorem, what he is after is an exact formula!

(Lecture given in Seattle in August 1996, on the occasion of the 100th anniversary of the proof of the prime number theorem, Atle Selberg comments about Riemann’s paper: A. Selberg, The history of the prime number theorem, A SYMPOSIUM on the Riemann Hypothesis, Seattle, Washington)

This exact formula has been obtained here: the five element subdivisions of the interval lead directly and precisely to the values of the zeta zeros, to the nth decimal precision desired.

The Riemann-Siegel formula is a local expression, while the Five Element Subdivision algorithm is a global formula.

It involves no transcendental or algebraic functions, but only the four elementary operations of mathematics.
 

Thus, all of the zeros of the zeta function must be located on the 1/2 critical line: if any of the Riemann zeta function ζ(s) non-trivial zeros would lay off the critical line, s = σ + it, σ = 1/2 - ε, then the values of all of the other zeros would have to be modified as well, all the way to the first zero, 14.134725.

The sum of any two sides of a triangle is greater than the third side.
 

mNgiI.jpg

 

The five elements sequence of proportions would be disrupted as the distance from the previous zero to the zero which is off the critical line, and from the zeta zero which finds itself on the σ = 1/2 - ε line to the next zero would be greater than the distances from that previous zero to the next two zeta zeros to be found on the critical line.

Moreover, since there are two counter-propagating zeta function waves, there would have be to TWO zeros off the critical line within the same 63.6363... segment.

To see the issues involved, here are the first five element subdivisions (first upper and lower bounds) for the second and third zeta zeros.

21.022

14.134725 + 6.3636 = 20.4975

14.134725 + 6.3636 + 0.80886 = 21.30656

(14.134725 + 63.6363) - 16.1773 - 12.066 - 8.998 - 6.7106 - 5.0045 - 3.732 - 2.7834 = 22.29945

(14.134725 + 63.6363) - 16.1773 - 12.066 - 8.998 - 6.7106 - 5.0045 - 3.732 - 2.7834 - 2.0757 = 20.22945

25.0108

14.134725 + 9.5445 + 1.68632 = 25.36602

14.134725 + 9.5445 + 0.99492 = 24.67

(14.134725 + 63.6363) - 16.1773 - 12.066 - 8.998 - 6.7106 - 5.0045 - 3.732 = 25.099425

(14.134725 + 63.6363) - 16.1773 - 12.066 - 8.998 - 6.7106 - 5.0045 - 3.732 - 1.64 = 23.459

Even the slightest deviation from the 21.022039639 and 25.010857580 values would invalidate the entire five element subdivision algorithm.

The values of the zeta zeros are a consequence of the precise five element subdivisions fractal.

Thus, if a zero should be located off the critical line, it would mean that all of the values of the previous zeros would have be modified as well, all the way to the first zero which is 14.134725.

 

 

Currently, the values of the zeta zeros are thought to be totally random:

http://math.sun.ac.za/wp-content/uploads/2011/03/Bruce-Bartlett-Random-matrices-and-the-Riemann-zeros.pdf

https://pdfs.semanticscholar.org/fc82/c1f7e35f23eb1695b0c78830c366e1258c88.pdf

One of the best mathematicians in the world, Dr. Yuri Matiyasevich (who solved Hilbert's tenth problem), could not find a scientific journal which would publish his results which prove that there is definite relationship between the values of the zeta zeros:

https://www.researchgate.net/publication/265478581_An_artless_method_for_calculating_approximate_values_of_zeros_of_Riemann's_zeta_function

https://phys.org/news/2012-11-supercomputing-superproblem-journey-pure-mathematics.html

 

The five element subdivision algorithm (fractal) creates the zeta zeros, which in turn are related to the distribution of the prime numbers.

"These zeros did not appear to be scattered at random. Riemann's calculations indicated that they were lining up as if along some mystical ley line running through the landscape." 

The mystical ley line has been revealed here: it is the five element subdivision algorithm.

"Present an argument or formula which (even barely) predicts what the next prime number will be (in any given sequence of numbers)."

The relationship between log p and the values of the zeta zeros:

http://www.dam.brown.edu/people/mumford/blog/2014/RiemannZeta.html

The log-prime figures give oscillating terms whose discrete frequencies correspond to the true zeros of the zeta function. And this method can be extended to large primes.

Since we now know that the five element subdivision algorithm creates the actual zeta zeros values, then these values can be anticipated in a very precise fashion, thus making possible the prediction of the next prime number.
 

<N(T)> = T/2π(logT/2πe) + 7/8 

Let <N(T)> = n (value of an integer)

n - 7/8 = T/2π(logT/2πe)

http://mathworld.wolfram.com/LambertW-Function.html

The Lambert W function is the inverse of f(W) = WeW.

Main subdivision point =~ 2πe ⋅ eW[(n - 7/8)/e]

Mathematica software for the Lambert W function:

http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=ProductLog

 

Global algorithm relating to the Lehmer pairs

(corresponding <N(t)> values in the parentheses).

7005.0629 and 7005.10056

7004.0437 (6707.487)
7005.0629 (6708.626)
7005.10056 (6708.667)
7006.74 (6710.498)

That is, the Lehmer pair will be located between the average number of zeta zeros values of 6708 and 6709.

Sacred cubit interval: 63.6363636363...

6999.999
7063.6363


For the first zeta function the values are:

7003.1814
7003.9903
7004.593
7005.0435
7005.17544
7005.379

The calculations for the second zeta function:

7004.541
7005.175
7005.48

Since 7005.175 and 7005.17544 are very close figures, a Lehmer pair must be located in the vicinity of these values.


Using 200/π = 63.66197724 as a sacred cubit interval the results are not as impressive: the nearest values are 7004.6825 and 7004.689, 7005.2652 and 7005.23.


63137.2115 and 63137.2324

63136.537 (82551.023)
63137.2115 (82552.013)
63137.2324 (82552.0434)
63138.2238 (82553.4973)

Sacred cubit interval: 63.6363636363...

63127.21
63190.846


For the first zeta function the values are:

63136.755
63137.42
63137.0866

(+9.5445, +0.66328, +0.33164)

The calculations for the second zeta function lead to these values:

63138.1574
63137.61
63137.063

(...-3.7322, -0.5474, -1.09478)

Since 63137.0866 and 63137.063 are very close figures, a Lehmer pair must be located in the vicinity of these values.

Using 200/π = 63.66197724 as a sacred cubit interval the results are not as impressive; the nearest values are: 63137.32 and 63137.185.


71732.9012 and 71732.91591

71732.02 (95246.674)
71732.9012 (95247.984)
71732.91591 (95248.00608)
71734.097 (95249.76)

Sacred cubit interval: 63.6363636363...

71718.11
71781.75


For the first zeta function the values are:

71734.287
71727.655
71729.341
71730.6
71731.536
71732.235
71732.757

The calculations for the second zeta function:

71732.7936
71732.06
71731.326
71730.6

Two pairs of zeta zeros which are very close: 71730.6 and 71732.757 and 71732.7936.

To distinguish between these choices the second five element subdivision algorithm will be applied.

53.4
106.8
136.1
160
534

63.636363
19.091
16.1773
12.7272
6.363


63.63 - 19.091 = 44.5453

44.5453
13.363
11.3252
8.91
4.454

44.5453 - 13.363 = 31.1823

31.1823
9.3547
7.9278
6.23646
3.11823

31.1823 - 9.3547 = 21.8276

21.8276
6.5483
5.5494
4.36552
2.18276

21.8276 - 6.5483 = 15.2793

15.2793
4.5838
3.88461
3.05586
1.52793

For the first zeta function, the values are:

71737.201
71730.8372
71731.8722
71734.2873
71732.6
71732.938
71732.77

The calculations for the second zeta function:

71733.4
71731.865
71730.337
71732.935
71732.663
71732.721

Since 71730.8372 and 71730.337 are not as close to one another as the corresponding pair using the first five element subdivision algorithm, it means we are not dealing with a Lehmer pair; the same analysis applies to 71731.8722 and 71731.865 (the corresponding pair using the five element subdivision algorithm are not this close to one another).

Amazingly, the two five element subdivision algorithms have located the precise interval of the Lehmer pair: 71732.757 and 71732.7936, 71732.938 and 71732.935.

The actual values are: 71732.9012 and 71732.91591.

Using 200/π = 63.66197724 as a sacred cubit interval the results are not as impressive; the nearest values are: 71732.9171 and 71732.909, after a long series of calculations (more involved than using 63.6363636 as a sacred cubit interval).


220538.853 and 220538.8702

220537.0585 (332251.37)
220537.4266 (332251.98)
220538.853 (332254.36)
220538.8702 (332254.39)
220539.8528 (332256.0258)

220538.853 is the 332254th zero, 220538.8702 is the 332255th zero, where the average spacing is 0.6.

Sacred cubit interval: 63.6363636363...

220499.78
220563.416


For the first zeta function the values are:

220537.0213
220538.341
220538.926
220538.676
220538.824

The calculations for the second zeta function:

220537.925
220538.863


Since 220538.824 and 220538.863 are very close figures, a Lehmer pair must be located in the vicinity of these values.  

Using the second five element subdivision algorithm, the following results are obtained:

220538.471
220538.901
220538.81
220538.64

220535.415
220539.871
220538.534
220538.738
220538.98

Again, the Lehmer pair must be located very close to the value of 220538.9.

Using 200/π = 63.66197724 as a sacred cubit interval the results are not as impressive: the nearest values are: 220538.8085 and 220538.8266.


435852.8393 and 435852.8572

435851.967 (703890.467)
435852.8393 (703892.015)
435852.8572 (703892.046)
435853.455 (703893.107)

Sacred cubit interval: 63.6363636363...

435845.45
435909.0865


435851.814
435852.623
435852.743

435853.6145
435852.7981

The Lehmer pair must be located around the value of 435852.78.


555136.9163 and 555136.9315

555136.284 (917905.02)
555136.9163 (917906.17)
555136.9315 (917906.195)
555137.412 (917907.066)

Sacred cubit interval: 63.63636363...

555099.9944
555163.631


555133.547
555137.2357
555136.385
555136.6
555136.763
555136.8831

555135.3877
555140.3352
555137.44
555136.92
555136.767


773657.1461 and 773657.1559

773656.6413
773657.1461
773657.1559
773658.041

Sacred cubit interval: 63.63636363

773627.265
773.690.9014


773655.51
773.657.278
773657.3665

773657.35


947107.8201 and 947107.8325

947107.2485
947107.8201
947107.8325
947108.2566

Sacred cubit interval: 63.63636363

947099.9905
947163.627


947106.354
947107.163
947107.766

947108.155
947107.7468

Both sets of five element subdivision algorithms are needed to detect the Lehmer pairs, the zeta zeros values which are most difficult to find.

 


http://www.dtc.umn.edu/~odlyzko/doc/zeta.derivative.pdf

1.30664344087942265202071895041619 x 1022

1.30664344087942265202071898265199 x 1022


The average spacing of zeros at that height is 0.128, while the above Lehmer pair of zeros is separated by 0.00032 (1/400th times the average spacing).

Using a very large number calculator:

205329683587299385104.8399385104 = 13066434408794226520207/63.63636363

12935770065999861261552 =  205329683587299385104 x 63

130664342794365258601.54936752 = 205329683587299385104 x .63636363
 
53.45063194549 = 0.8399835 x 63.63636363


13,066,434,408,794,226,520,153.54936752
13,066,434,408,794,226,520,217.18573115


The calculations for the first zeta function:

13,066,434,408,794,226,520,169.72670085
(+16.17733333)

13,066,434,408,794,226,520,181.79268471
(+12.06598386)

13,066,434,408,794,226,520,190.791012836
(+8.998328126)

13,066,434,408,794,226,520,197.501606019
(+6.710593183)

13,066,434,408,794,226,520,202.506097991
(+5.004491972)

13,066,434,408,794,226,520,206.238247924
(+3.732149933)

13,066,434,408,794,226,520,207.332996246
(+1.094748322)

13,066,434,408,794,226,520,206.785622085
(+0.547374161)

13,066,434,408,794,226,520,206.924786491
(+0.139164406)

13,066,434,408,794,226,520,207.028569738
(+0.103783247)

13,066,434,408,794,226,520,207.105967138
(+0.0773974)

13,066,434,408,794,226,520,207.163687018
(+0.05771988)

13,066,434,408,794,226,520,207.189083398
(+0.02539638)


The computations for the second zeta function:

13,066,434,408,794,226,520,207.64027661
(-9.54545454)

13,066,434,408,794,226,520,207.308682645
(-0.331593965)

13,066,434,408,794,226,520,207.224378195
(-0.08430445)

0.247289515 interval

13,066,434,408,794,226,520,207.187284768
(-0.037093427)


The second five element subdivision algorithm:

13,066,434,408,794,226,520,172.64027661
(+19.090909)

13,066,434,408,794,226,520,186.00391297
(+13.363636)

13,066,434,408,794,226,520,195.35845842
(+9.35454545)

13,066,434,408,794,226,520,201.906640238
(+6.548181818)

13,066,434,408,794,226,520,206.490367508
(+4.58372727)

1.06953636
0.320860909
0.271919
0.21390727
0.106953636

13,066,434,408,794,226,520,206.811228417
(+0.320860909)

13,066,434,408,794,226,520,207.035831017
(+0.2246026)

13,066,434,408,794,226,520,207.193052861
(+0.1572218844)


13,066,434,408,794,226,520,210.82209485
(-6.363636)

13,066,434,408,794,226,520,208.91300395
(-1.9090909)

13,066,434,408,794,226,520,207.576640314
(-1.336363636)

3.11818181

13,066,434,408,794,226,520,207.264822133
(-0.3118181)

13,066,434,408,794,226,520,207.202458497
(-0.062363636)

0.07927665
0.0623636

0.01691303
0.00507391

13,066,434,408,794,226,520,207.197384587
(-0.00507391)

0.0118391
0.003551737

13,066,434,408,794,226,520,207.19383285
(-0.0035517370
 

7954022502373.43289015387
7954022502373.43289494012

t2 - t1 = 4.7863 x 10-6 = 0.0000047863

7,954,022,502,331.87047696
7,954,022,502,395.50684059


7,954,022,502,348.04781029
(+16.17733333)

7,954,022,502,360.11379415
(+12.06598386)

7,954,022,502,369.112122276
(+8.998328126)

7,954,022,502,373.071330025
(+3.959207749)

2.751385438
0.69951223
0.412707815
0.2751385438
0.13756927

7,954,022,502,373.3464685688
(+0.2751385438)

0.13756927
0.03497561

7,954,022,502,373.3814441788
(+0.03497561)


0.03497561 = 1/28.5914, where 286.1 = 450 sc (1 sacred cubit = 2/π or 7/22)

0.10259366
0.0260834

7,954,022,502,373.4075275788
(+0.0260834)

0.07651025
0.019451965

7,954,022,502,373.4269795438
(+0.019451965)

0.057058282

0.0057058282

7,954,022,502,373.432685372
(+0.0057058282)


7,954,022,502,395.50684059

7,954,022,502,379.32950729
(-16.1773333)

7,954,022,502,374.58360426
(-4.74590303)

2.372951517
0.60329919

7,954,022,502,373.98030507
(-0.60329919)


7,954,022,502,373.530388664
(-0.449916406)

1.319735916
0.131973591
0.065986795

7,954,022,502,373.464401869
(-0.065986795)

0.065986795
0.016776482

7,954,022,502,373.447625387
(-0.016776482)

0.04921031
0.01251123

7,954,022,502,373.435114157
(-0.01251123)

0.036699082

7,954,022,502,373.43327920286
(-0.00183495414)

7,954,022,502,373.43281268412
(-0.00046651874)


The second five element subdivision algorithm calculations:

7,954,022,502,350.96138605
(+19.09090909)

7,954,022,502,364.32502241
(+13.36363636)

7,954,022,502,373.67956786
(+9.35454545)



7,954,022,502,376.4159315
(-19.090909)

4.45454545
1.3363636

7,954,022,502,375.0795679
(-1.3363636)

7,954,022,502,374.1441134
(-0.9354545)

2.18272727
0.65481818
0.5549366
0.43654545
0.2182727

7,954,022,502,373.48929522
(-0.65481818)


Using the Riemann-Siegel asymptotic formula, the sum will feature at least O(4.56 x 1010) terms (for t = 1.30664344087942265202071895041619 x 1022):

siegel1.png

Using the five element subdivision algorithms, we only need to translate/shift the 63.63636363 interval by a factor of k: k = [t/63.6363636363] x 63.6363636363, where [ x ] denotes the integer part, and then simply apply the five element partition process for the two zeta functions to detect both regular zeta zeros and Lehmer pairs, carefully keeping a check on the average number of zeros values which are close to an integer (these are the values which are equivalent to a five element subdivision figure). 

 

The zeta zeros are generated by the five element subdivision algorithm. These zeros, in turn, determine the distribution of the prime numbers.

Mathematicians have concentrated for far too long on the RH, and have neglected the more important issues: what do these zeros actually represent? Is there a hidden pattern to these values? 
 

"The lack of a proof of the Riemann hypothesis doesn't just mean we don't know all the zeros are on the line x = 1/2 , it means that despite all the zeros we know of lying neatly and precisely smack bang on the line x = 1/2 , no one knows why any of them do, for if we had a definitive reason why the first zero 1/2 + 14.13472514 i has real value precisely 1/2 we would have a reason to know why they all do. Neither do we know why the imaginary parts have the values they do.

Answers to such questions depend on a much more detailed knowledge of the distribution of zeros of the zeta function than is given by the RH. Relatively little work has been devoted to the precise distribution of the zeros."

C. King

Edited by sandokhan

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https://arxiv.org/pdf/1704.05834.pdf

On large gaps between zeros of L-functions from branches

Andre LeClair (Cornell University) proves that the normalized gaps between consecutive ordinates tn of the zeros of the Riemann zeta function on the critical line cannot be arbitrarily large.

 


https://arxiv.org/pdf/1508.05870.pdf

Lehmer pairs revisited

The Riemann hypothesis means that the de Bruijn-Newman constant is zero.

Unusually close pairs of zeros of the Riemann zeta function, the Lehmer pairs, can be used to give lower bounds on Λ.

Soundararajan’s Conjecture B implies the existence of infinitely many strong Lehmer pairs, and thus, that the de Bruijn-Newman constant Λ is 0.

 

altz.jpg

 

The factor (1 - 2^[1 - s])^(-1) determines the exact locations of the zeta zeros, especially those of the Lehmer pairs.

2π/ln2 x 1 = 9.064720284
2π/ln2 x 2 = 18.129440568
2π/ln2 x 3 = 27.194160852
2π/ln2 x 4 = 36.258881136
2π/ln2 x 5 = 45.32360142
2π/ln2 x 6 = 54.388321704
2π/ln2 x 7 = 63.453041988
2π/ln2 x 8 = 72.517762272
2π/ln2 x 9 = 81.582482556
2π/ln2 x 10 = 90.64720284
2π/ln2 x 11 = 99.711923124
2π/ln2 x 12 = 108.776643408
2π/ln2 x 13 = 117.841363692
2π/ln2 x 14 = 126.906083976
2π/ln2 x 15 = 135.97080426
2π/ln2 x 16 = 145.035524544

(an addition by the factor of 1 to the decimal part after the multiplication by 16)

(k/2)(2π/ln2)

The zeta zeros cannot be generated at the odd or even integer values of k; the zeta zeros can be generated only at the fractional values of k.

This is the starting point of the so-called "excess of nines = 1" theory as it applies to the Lehmer pairs.

"It is evident that all even or odd (whole numbers) values of k produce an excess of nines=1 and therefore cannot generate a zeta function zero. Further, it is true that all zeros occur from the fractional values of the k's; when an unscheduled "Excess of Nines=1" occurs, so does a Lehmer event.The plotted data briefly passes through an excess of nines=1, wavers, then becomes fractional again, crosses the real axis and produces a zero."

While these facts do explain the occurrence of the regular Lehmer pairs, a deeper explanation is needed to account for the existence of the strong (high quality) Lehmer pairs.


http://www.dtc.umn.edu/~odlyzko/doc/zeta.derivative.pdf

1.30664344087942265202071895041619 x 1022

1.30664344087942265202071898265199 x 1022


The average spacing of zeros at that height is 0.128, while the above Lehmer pair of zeros is separated by 0.00032 (1/400th times the average spacing).
 

So, the pattern of these strong Lehmer pairs has to be revealed/deciphered.

 

Highest zeta zero ever computed: 
t ≈ 81029194732694548890047854481676712.9879 ( n = 1036 + 4242063737401796198).

https://arxiv.org/pdf/1607.00709.pdf

1273315917355388788579148020712834 x 63 = 80218902793389493680486325304908542

1273315917355388788579148020712834 x 0.63636363 = 810291939305055209561529176768134.23182742


81,029,194,732,694,548,890,047,854,481,676,676.23182742
81,029,194,732,694,548,890,047,854,481,676,739.86819105


81,029,194,732,694,548,890,047,854,481,676,692.40916072
(+16.1773333)

81,029,194,732,694,548,890,047,854,481,676,704.47514472
(+12.065984)

81,029,194,732,694,548,890,047,854,481,676,713.47347282
(+8.9983281)

81,029,194,732,694,548,890,047,854,481,676,709.78410162
(+5.3089569)

81,029,194,732,694,548,890,047,854,481,676,710.72208732
(+0.9379857)

81,029,194,732,694,548,890,047,854,481,676,711.42159955
(+0.69951223)

81,029,194,732,694,548,890,047,854,481,676,711.943267789
(+0.521668239)

81,029,194,732,694,548,890,047,854,481,676,712.332307089
(+0.3890393)

81,029,194,732,694,548,890,047,854,481,676,712.622437039
(+0.29012995)

81,029,194,732,694,548,890,047,854,481,676,712.838804339
(+0.2163673)



81,029,194,732,694,548,890,047,854,481,676,723.69085805
(-16.177333)

81,029,194,732,694,548,890,047,854,481,676,711.62487405
(-12.065984)

81,029,194,732,694,548,890,047,854,481,676,716.5720035
(-7.11885455)

81,029,194,732,694,548,890,047,854,481,676,715.3142453
(-1.2577582)

81,029,194,732,694,548,890,047,854,481,676,714.37625955
(-0.93798575)

81,029,194,732,694,548,890,047,854,481,676,713.6767473
(-0.69951225)

81,029,194,732,694,548,890,047,854,481,676,713.15507905
(-0.52166825)

 

s = r x θ

r = 68.1 (136.2/2, 22.7 = 1.362 x 16.66666, 38.136 = 1.362 x 28, 51.756 = 1.362 x 38, 68.1 = 1.362 x 50, 81.72 = 1.362 x 60, 98.064 = 1.362 x 72, 118.494 = 1.362 x 87)

θ = 136.12° 

sin 136.12° = ln2

136.12° = 2.375742 radians

s = 161.78804

63.6363/16.1773 = 1/0.25422

2π/ln2 = (10s - 1000)/r

10 x 136.12° radians - 1000/r = 2π/ln2

136.12° radians x 3.819072 = 2π/ln2

That is, 2π/ln2 is the arclength corresponding to the 136.12° expressed in radians multiplied by 6 sacred cubits.

 

 

Edited by sandokhan

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There is another very interesting phenomenon: very close double Lehmer pairs.

http://www.slideshare.net/MatthewKehoe1/riemanntex (pg. 64-87)

441296.9992 / 441297.0149

and

441649.8183 / 441649.8273

506616.5065 / 506616.5305

and

506959.3064 / 506959.327

675064.2749 / 675064.2909

and

675149.609 / 675149.6213

692620.9588 / 692620.9806

and

692736.741 / 692736.7631

847172.8025 / 847172.8171

and

847263.1402 / 847263.1502

1055407.79 / 1055407.813

and

1055657.193 / 1055657.216

1438925.829 / 1438925.849

and

1439457.546 / 1439457.567

1579400.943 / 1579400.968

and

1579721.076 / 1579721.097

1662448.536 / 1662448.546

and

1662515.735 / 1662515.743

Even on a larger scale, this phenomenon still can be observed.

18580341990011.15934

and

18523741991636.36437

 

Since 2π/ln2 x 15 = 135.9708043... is the value of the "unscheduled excess of nines=1" Lehmer pairs occurrence, we can use it as another shortcut formula for finding both regular and strong Lehmer pairs.

(2π/ln2 x 15) x k = 135.9708043 x k

The first strong Lehmer pair is:

415.0188 and 415.455 

2π/ln2 x 15 x 3 = 407.9124

The most famous Lehmer pair is of course:

17143.786 and 17143.8218

leh3.jpg

2π/ln2 x 15 x 126 = 17132.321

 

35615956517.47854

2π/ln2 x 15 x 261938273 = 35,615,956,530.4284

 

2414113624163.41943

2π/ln2 x 15 x 17754647499 = 2,414,113,624,157.0292

 

7954022502373.43289015387

2π/ln2 x 15 x 58498019445 = 7,954,022,502,352.206

 

13066434408794226520207.1895041619

2π/ln2 x 15 x 96097356261743157503 = 13,066,434,408,794,226,520,208.9124

 

So, this shortcut formula is able to detect some of the best known strong Lehmer pairs.

However, this is still not enough. We need to know/discover the hidden pattern of the strong Lehmer pairs from a basic arithmetical point of view.

 

The average number of zeta zeros on the entire critical strip:

N(T) = (T/2π)(lnT/2π) - T/2π 

T = 40

Let T1 = T + L(T)

L(T) = average spacing = 2π/log(t/2π)

N(T1) = (T + L(T))/2π {ln[(T + L(T)) ⋅ /2π] - 1} = N1

L(T) = 3.3945 

T1 = 43.3945

N = 5.41765

N1 = 6.44

Zeta zeros

37.586 = z1
40.9187 = z2

8(N + N1) - 2z1 = 19.6892

8(N + N1) - 2z2 = 13.0238

8(N + N1) - z1 - z2 = 16.3565

16N - z1 - z2 = 8.1777

16N1 - z1 - z2 = 24.5353


8(N + N1) - z1 - z2 = 2 x (16N - z1 - z2) = 32N - 2z1 - 2z2

z1 + z2 = 32N - 8(N + N1)

Edited by sandokhan

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theorem.jpg

https://arxiv.org/pdf/1612.08627.pdf

Few mathematicians who study the zeta function remember or have knowledge of the fact that D.H. Lehmer proved the existence of an infinite number of Lehmer pairs:

https://projecteuclid.org/download/pdf_1/euclid.acta/1485892173

Lehmer, D. H. On the roots of the Riemann zeta-function. Acta Math. 95 (1956), 291--298

lehmer1.jpg

H.M. Edwards acknowledges this proof in his treatise on the zeta function (Riemann's Zeta Function, section 8.3, pg 179):

lehmer2.jpg

The Riemann hypothesis means that the de Bruijn-Newman constant is zero.

The existence of infinitely many Lehmer pairs proves that the de Bruijn-Newman constant Λ is 0.

An infinite sequence of such Lehmer pairs is given by the formula:

(636.3 x 3n - 16.9 x n)2π/ln2

(n = 1,2,3...)

n = 1

17150.45078


http://www.dtc.umn.edu/~odlyzko/zeta_tables/zeros1

17143.786536184
17143.821843505

n = 2

34300.90155

34295.104944255
34295.371984027

n = 3

51451.35233

51448.076349964
51448.729475327
51449.153911623

n = 4

68601.80311

68597.636479797
68597.971396943

n = 5

85752.25388

85748.621773488
85748.861163006

n = 6

102902.7047

102907.166732245
102907.475751344

n = 7

120053.1554

120055.446373211
120055.565321075

It can be checked even at much greater heights on the critical line:

http://www.dtc.umn.edu/~odlyzko/zeta_tables/zeros4

However, what is still needed is an understanding of the nature of the strong Lehmer pairs, how they relate to the two counterpropagating zeta functions.

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Global algorithm for the strong Lehmer pairs

 

π/ln2 is the sacred cubit (basic unit of distance) of the Lehmer pairs.

1400/22 is the sacred cubit distance for regular zeta function zeros.

Infinite sequence formula for all Lehmer pairs:

{n ⋅ 2π/ln2 + n ⋅ 2π/ln2 + π/ln2}/2 =  π/ln2 x (4n + 1)/2

{n ⋅ 2π/ln2 + π/ln2 + (n + 1) ⋅ 2π/ln2 }/2 =  π/ln2 x (4n + 3)/2

There are several choices for the optimum interval which can be used for the global subdivision algorithm to find the strong Lehmer pairs:

2π/ln2 x 10 = 90.6472...

2π/ln2 x 15 = 135.9708...

2π/ln2 x 75/2 = 75π/ln2 = 2.5 x 135.9708... = 339.9270106...

2π/ln2 x 100sc =  576.84583...

The best version is 75π/ln2 = 2.5 x 135.9708... = 339.9270106...

The subdivision proportions are as follows:

534 sc = 339.9270106...
160 sc = 101.81818...
135.9708 x sc = 86.52687538...
106.8 sc = 67.985402...
53.4 sc = 33.9927...

1018.1818 = 4 x 254.545454... = 16 x 63.636363...

For the 100sc interval the subdivision proportions were:

534
136.1 
80
53.4
26.7

and 

534
160
136.1
106.8
53.4

List of strong Lehmer pairs:

https://www.slideshare.net/MatthewKehoe1/riemanntex (pg 64-88)

The first such values are: 415.018809755 and 415.455214996, 7005.062866175 and 7005.100564674, 17143.786536184 and 17143.821843505, 23153.514967223 and 23153.574227077...

http://www.dtc.umn.edu/~odlyzko/zeta_tables/zeros1

All of the zeros of the zeta function are generated by the five elements subdivision algorithm, therefore the location of all of the Lehmer pairs (including the strong Lehmer pairs) must be related to the subdivision values, but on a larger scale. 

The same global algorithm successfully employed before to find the zeta zeros on a 100 sc interval, will be used again, featuring the two counterpropagating zeta functions.

33992.701
10181.818
8652.687
6798.54
3399.27

33992.701 - 10181.818 = 23810.883

23810.883
7132.0626
6060.952
4762.1766
2381.0883

23810.883 - 7132.0626 = 16678.82

16678.82
4995.8
4245.518
3335.764
1667.882

1667.882
499.58
424.5518
333.5764
166.7882


10181.818 + 7132.0626 = 17313.8806
10181.818 + 6060.952 = 16242.77

23810.883 - 6060.952 = 17749.931
23810.883 - 7132.0626 = 16678.82

7132.0626 - 6060.952 = 1071.1106

1071.1106
320.83
272.646
214.222
107.111

16242.77 + 320.83 = 16563.6

17749.931 - 320.83 = 17429.1

1071.1106 - 320.83 = 750.281

750.281
224.73
190.98
150.056
75.28

17429.1 - 224.73 = 17204.37

16563.6 + 224.73 = 16788.33

That is, upper and lower bounds are being obtained for the Lehmer pair located at 17143.786...

We also have a lower bound estimate for the Lehmer pair located at 7005.1..., 6798.54, and an upper bound for the Lehmer pair located at 23153.5..., 23810.883.

The Lehmer phenomenon, a pair of zeros which are extremely close, is related to the close proximity of some of the values of the two subdivisions of the 63.6363... segment.

In the same way, strong Lehmer pairs are related to the close proximity of some of the values of the two subdivisions of the 10n x 339.927106... segment. The same algorithm can be applied for the 339927.106... segment or for the 33992710.6... segment, however the calculations involving the two subdivision fractals will be more involved since now we have to obtain many more significant digits. 
 

On the extreme values of the zeta function:

https://www2.warwick.ac.uk/fac/sci/maths/people/staff/stefan_grosskinsky/rcssm/ws2/00ExtremeBehaviour.pdf

http://mat.uab.cat/~bac16/wp-content/uploads/2013/12/talk.bondarenko.pdf

https://heilbronn.ac.uk/wp-content/uploads/2016/05/Hughes_MaxZeta0T.pdf

http://siauliaims.su.lt/pdfai/2004/stesle-04.pdf

Edited by sandokhan

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