Need description of Prime# distribution in Riemann hypothesis

[Trurl]

Can anyone explain in one paragraph or equation how the Riemann hypothesis implies a pattern in Prime numbers?

The hypothesis an easy enough to follow the problem, but the pattern in Primes is not clear to me.

I’m sure a Google search may be helpful, but I wanted to work it through on my own. I find it helpful to take one small part if the problem and see what it does. Take the book Practical Cryptography. It explains all the ciphers. They are all open source. Doesn’t mean you can solve them but it lets you see the inner workings.

That is what I need an explanation of the patterns in Primes in the Riemann hypothesis. I hope you understand why I just don’t google it. I want to search for nontrivial zeros. And I want to keep it simple.

[Trurl]

Can no one explain the distribution of Primes according to the Riemann hypothesis?

 

All that I have seen or read is that the importance of the Riemann hypothesis is the distribution of Primes, but no one explains why it works.

 

This is different from my last project. The reason I want to know is the same though.

 

Is it a correct statement that:

                *the Prime Factorization problem (RSA)

                *the discrete logarithmic problem

                *elliptic curve cryptography

All rely on factorization and Prime distribution?

 

 

My goal is to start a different math project. I just want to find zeros and start with zeros that already exist.

 

In my post “Simple Yet Interesting” someone asked why I was factoring semi-Primes to hopefully solve RSA. As most of you know, that is the basis of RSA’s one-way-function. That was what the cipher was based. All I tried to do was take the semi-Prime and see why it couldn’t be factored. I want to take the simplest concept of the Riemann hypothesis and see why it is I cannot find a zero.

 

I am going to attach a paper I wrote 11 years ago. It was a simple outline. My instructor corrected grammar and citation format. I argue that it is a good outline of cryptography for only having to be a 3 page paper. Let me know what you think. Is it that bad? I get frustrated when I research something and the comments the instructor leaves on my paper is, “the comma comes after the parenthesis.” Is the paper a good description of how cryptography evolved?

TrurlCryptoSurveySFN002.docx

[Trurl]

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

 

 

https://mathworld.wolfram.com/RiemannZetaFunctionZeros.html

 

https://m.youtube.com/watch?v=sD0NjbwqlYw&pp=QAFIAQ%3D%3D

 

 

https://m.youtube.com/watch?v=d6c6uIyieoo

 

 

 

These are the best explanations I have found so far. It is enough to get started. I still don’t understand the displacement of Prime numbers as it relates to the zeta function.

 

My attempts in my post “Simple Yet Interesting” I encountered complex numbers and I also had decimals and fractions. I am looking back here the Riemann Zeta function also deals with those. So I’m wondering why a was searching for a perfect integer.

 

I have some trials I want to test for zeros. The complex numbers are a game changer.

 

 

Well I moved on to a new problem. Unfortunately it too is related to Prime numbers. Proving it wrong I would need to find a nontrivial zero. I don’t have a clue how to prove it is always true. If that is even possible. My goal is to learn and perform my own zeros tests.

[Trurl]

This is just first inspection. But a kind of intuition starting point. Thus it probably doesn't work. But I just needed a starting point to test where I think zeros could occur. And where if they didn't occur would show. I believe zeros could only occur with even numbered fractions. But then again what do I know.

 

In memory of Pappy Craylar:

 

Hypothesis on critical zeros on Riemann Zeta Function

Non-critical zeros have been found at an x-value of ½  Possibly forming an isosceles triangle. What if we were to expand this to all even powers. That is ½ , ¼ , ⅙ … towards zero and ½ + ½^(½^n)... towards 1?

This fraction combined with and complex number would alternate between positive and negative values cancelling out to form zero.

First we should test values at ¼ that are a similar isosceles triangle to ½ non-trivial zeros. If we can not find a trivial zero at these values it supports the Riemann Hypothesis.

[Trurl]

Consider these problems:

 

An odometer turns so that the first numbers equal the last. For example, 096096 has 3 numbers that equal. But can you tell for which numbers this works for. For instance, if you had 123123 it would have 3 equal numbers, but it would take longer to get that size number. If 012012 occurred how close would it be to 021021. Would smaller numbers be closer of further away from similar patterns of the same series of numbers.

 

Obviously, you could argue that because we are going through all real numbers the occurrence of numbers and patterns are equal. The pattern is simple. But going linearly (counting) makes this observation difficult to see. And that is how I feel with patterns in Prime numbers. Most attempts at finding a pattern are linearly starting at 1 and counting. But I hypothesis that is why no pattern is found.

 

Problem 2:

We all leant that a modulus can be described by a clock. But what if you had 2 or more clocks that were rotating at different times at different rates of speed of rotation. Would they every have the exact time? That is rotating to infinity. And if they never do have the same time, could you prove it mathematically?

 

 

These are just some questions I had. I found a book on the Riemann Hypothesis that is geared to the undergraduate. But it is better is it easy to read so that I can better understand it. I heard in a podcast that an Indian mathematician claimed to have proved Riemann’s work. He posted it on the Internet. It has been years, and no one can make any sense of it. That is why it is best to keep it simple. You could spend years studying the Riemann Hypothesis and never know what Riemann knew. There is also a rumor that after Riemann passed away his maid took most of his papers on his desk and burnt them. She never explained why she did that. So, a pattern of Primes could have been solved. Instead, we are looking at what was published. I’m just looking for simple ways to understand it.

[Genady]

7 hours ago, Trurl said:

An odometer turns so that the first numbers equal the last. For example, 096096 has 3 numbers that equal. But can you tell for which numbers this works for. For instance, if you had 123123 it would have 3 equal numbers, but it would take longer to get that size number. If 012012 occurred how close would it be to 021021. Would smaller numbers be closer of further away from similar patterns of the same series of numbers.

Take one of such numbers, say, ABCABC. The next one is AB(C+1)AB(C+1). Take a difference:

AB(C+1)AB(C+1) - ABCABC = AB(C+1)*1000 + AB(C+1) - ABC*1000 - ABC = 1000 + 1 = 1001

So, they appear every 1001 numbers regardless of being smaller or larger.

[Trurl]

Awesome now we have a formula.

 

But what is we counted all different digits for 3 combinations and wanted to find how many occur between any given number. Say every 1001 units. I know they will all occur between 000000 and 999999 but what if you say 064064 and 065065 are 1001 distance apart. But 6400640 is significantly larger.

If you chose a range are there more combinations occurring linearly from say 000000 to 030000 than 050000 to 080000?

The combinations all occur over the range of real numbers, but do they occur evenly over an equal distance from zero? In other words, “is this pattern evenly distributed over the linear odometer? That is as they occur “counting.” (The odometer is just counting.)

I have been working with this for quite some time. When my odometer turned 096096, I thought it was a reason to share and see if this meant something.

019019 091091 901901

 

092092 029029 209209

[Genady]

The next after 640640 is 641641. 1001 apart.

There are 30000/1001 between 000000 and 030000 as well as between 050000 and 080000. Yes, linearly, evenly.

What is your question?

[NTuft]

Hi Trurl,
 

On 2/28/2022 at 8:58 PM, Trurl said:

Can anyone explain in one paragraph or equation how the Riemann hypothesis implies a pattern in Prime numbers?

The hypothesis an easy enough to follow the problem, but the pattern in Primes is not clear to me.

I’m sure a Google search may be helpful, but I wanted to work it through on my own. I find it helpful to take one small part if the problem and see what it does. Take the book Practical Cryptography. It explains all the ciphers. They are all open source. Doesn’t mean you can solve them but it lets you see the inner workings.

That is what I need an explanation of the patterns in Primes in the Riemann hypothesis. I hope you understand why I just don’t google it. I want to search for nontrivial zeros. And I want to keep it simple.

I found this slideshow presentation informative Prime numbers and the Riemann zeta function, Carl Wang-Erickson, Nov. 12, 2019:

Quote

[...]

For primes up to a real number A, what is the average size of the gap between primes?

Stated differently: Of the ∼ A numbers up to a real number A, what proportion of them are prime?

let’s define

π(X ) := the number of primes ≤ X
We call this the prime counting function.
How are the primes distributed on average?

“Average size of gaps up to A” = A/π(A)

“Proportion of numbers up to A that are prime” = π(A)/A

We find that the function

X/log X

is a good approximation for π(X). (The base of “log” is e.) [Ed.: i.e. ln, natural logarithm base]

We propose that π(X) behaves like X/ log X as X gets large. This means that:

the average gap between primes up to A is about log A.

the proportion of numbers up to A that are prime is about 1/ log A.

There is some difficult math, but if you go through the presentation I think it can clear up how the Riemann hypothesis points at a pattern in prime number's average, logarithmic distribution.

On 3/23/2023 at 9:10 PM, Trurl said:

 

[...]Obviously, you could argue that because we are going through all real [Ed.: natural] numbers [...]

 

Problem 2:

We all leant that a modulus can be described by a clock. [...]

I would like you to describe that to me. I think of the argument (angle) for a complex number as rotating around.

On 3/23/2023 at 9:10 PM, Trurl said:

 

 

[...] I found a book on the Riemann Hypothesis that is geared to the undergraduate. But it is better is it easy to read so that I can better understand it. I heard in a podcast that an Indian mathematician claimed to have proved Riemann’s work. He posted it on the Internet. It has been years, and no one can make any sense of it. [...] I’m just looking for simple ways to understand it.

Can you refer me to that document or paper, please?

On 3/25/2023 at 4:20 PM, Trurl said:

 

Awesome now we have a formula. [Ed.: agreed, that is interesting +1 Genady]

[...] over the range of real [Ed.:natural] numbers, [...]

I haven't gone through links or the paper you've provided, and I don't much understand the odometer question, but I read you're working at cryptography applications, so I'll think about it.

[Trurl]

Thanks NTuff. That presentation is excellent. It started off simple and you see how complex it gets. It defines the problem in a short series of slides. And that is what is important: understanding the problem. Wikipedia’s description is hard to understand.

 

I asked Chat gpt if an equation that equals zero at the Prime numbers could relate to the Zeta function. It said no because the Zeta function determines distribution. It said the Mangoldt function would relate better. This may or may not be true, because the presentation showed how the Mangoldt function relates to the Zeta function. So how does the equation I put in “Simple Yet Interesting” compare to Mangoldt?

 

BTW the equation for the odometer is not the one I am comparing to the Zeta function. It is just a model I used to picture patterns as I drive. But I think the problem of finding the distribution of repeating numbers in the odometer relates to the problem of the distribution of Primes.

 

The picture of the podcast is where I heard that an Indian mathematician uploaded his proof of the Riemann hypothesis and no one can tell if he is right.
 

[Trurl]

Link to 2013 paper:

 

https://annals.math.princeton.edu/wp-content/uploads/annals-v179-n3-p07-s.pdf

 

[NTuft]

On 3/30/2023 at 7:25 PM, Trurl said:

Link to 2013 paper:

 

https://annals.math.princeton.edu/wp-content/uploads/annals-v179-n3-p07-s.pdf

 

Thanks, but that is the famously well-received paper by Yitang Zhang, which earned him a university position: Unheralded Mathematician Bridges the Prime Gap . See here for a new recent result from Zhang,  Mathematician who solved prime-number riddle claims new breakthrough:

Quote

"After shocking the mathematics community with a major result in 2013, Yitang Zhang now says he has solved an analogue of the celebrated Riemann hypothesis."

He is Chinese-American. Whoever was mentioned as an Indian mathematician publishing on Riemann's hypothesis ca. 2013 is likely someone else--I may have to investigate what you're pointing to in the podcast to find out.

[Trurl]

The podcast is too vague. There are just too many people with papers on the Internet that claim to have solved the Riemann Hypothesis.

But if you are just searching for zeroes, are there any equations that you could just plug and chug numbers into? I have seen s^-1 summed. But I am searching for other ways to test for zeroes.

[Sensei]

21 minutes ago, Trurl said:

But I am searching for other ways to test for zeroes.

Learn programming..

 

[Trurl]

You’re right that my programming skills are lacking. But as with large Primes programming, large numbers in C++ are difficult.

In the Simple Yet Interesting, I graphed 2564855351 and it resulted in zero. That is not the same as the Riemann zeroes, but I want an educated guess to plug and chug. One mathematician said trillions of zeroes have been found at 1/2. But again I don’t know how to program imaginary numbers. I rely on Wolfram Alpha.

But I also want to program something different; something of mine. Should I stick with Mathematica? Also I need something to plug and chug. But I guess finding what to plug and chug is up to me. But what is a good starting point?

I treat working with Primes as a learning exercise. They are fifty math problems in one. But the solution may not even be possible. The computer can’t make a proof. But it can disprove if it isn’t at 1/2.

[Genady]

9 hours ago, Trurl said:

But I also want to program something different; something of mine. Should I stick with Mathematica? Also I need something to plug and chug. But I guess finding what to plug and chug is up to me. But what is a good starting point?

Programming is easy. To learn programming is easy. Start with any available language. I hear that Python is quite popular among scientists and mathematicians. But it really doesn't matter much. Eventually you learn and use programming languages as needed.

[Trurl]

You are right I need to work at becoming a better programmer. I don’t know much programming beyond what I learned in college. I tried to bridge the gap by using math scripting in Mathematica. It gave me access to things that require some more know how than my current level. And I am not against asking Chat gpt to program it for me. But remember, programming can find a counter example, but cannot prove math. I also am still looking for a better way to analyze graphs. I want the computer to analyze the graph according to math algorithms I input.

 

NTuff linked to a college presentation which referenced the book: Prime Numbers and the Riemann Hypothesis. I have been reading this book. The book makes the topics understandable, but the content is still complex. The is no easy way to put the Riemann Hypothesis, but this book explains the problem and does a good job at explaining the approaches to a solution.

 

Reading in this book, I have wondered if I have stair steps in my graphs. And if I should be concentrating on finding large Primes instead of finding a pattern.

 

I was looking at my odometer again today. It hit 097777 and then 097797. That is a short time to have 97 and 77 repeated. And it made me wonder is the reason we can’t find a pattern in Primes is because we are counting? 2, 3, 5 ,7 …. Is there a way to find Prime numbers in we don’t go by counting numbers? I know this is advanced and I don’t have a solution, but can you see what I mean? Does that statement make sense?

[Genady]

49 minutes ago, Trurl said:

 

You are right I need to work at becoming a better programmer. I don’t know much programming beyond what I learned in college. I tried to bridge the gap by using math scripting in Mathematica. It gave me access to things that require some more know how than my current level. And I am not against asking Chat gpt to program it for me. But remember, programming can find a counter example, but cannot prove math. I also am still looking for a better way to analyze graphs. I want the computer to analyze the graph according to math algorithms I input.

 

NTuff linked to a college presentation which referenced the book: Prime Numbers and the Riemann Hypothesis. I have been reading this book. The book makes the topics understandable, but the content is still complex. The is no easy way to put the Riemann Hypothesis, but this book explains the problem and does a good job at explaining the approaches to a solution.

 

Reading in this book, I have wondered if I have stair steps in my graphs. And if I should be concentrating on finding large Primes instead of finding a pattern.

 

I was looking at my odometer again today. It hit 097777 and then 097797. That is a short time to have 97 and 77 repeated. And it made me wonder is the reason we can’t find a pattern in Primes is because we are counting? 2, 3, 5 ,7 …. Is there a way to find Prime numbers in we don’t go by counting numbers? I know this is advanced and I don’t have a solution, but can you see what I mean? Does that statement make sense?

Is it this book? I have it for some time, but there always are other books in front of it somehow...

[Trurl]

That’s the book.

The authors wrote it working in just 1 week a year for 10 years. The one I ordered wasn’t too expensive. I think the book is printed as you order.

Excellent read. It puts the number theory into sentences. It is good for undergraduate or lower level graduate. That is my level. For me the notation of number theory is above my education. I know the math of things like the Riemann Hypothesis is complex, but the idea should still be generalized in words.

Didn’t someone say if you can’t explain the problem simply, you don’t understand it.

[Trurl]

So in 2007 I tried to make a logarithmic spiral show a pattern of Prime numbers.

 

Long story short, I couldn’t get it to work. It was before I heard of the Riemann Hypothesis. The area of the logarithmic spiral could be set to the integral of li(X).

 

I am not claiming it works. I am asking what you think of placing the Riemann Stair Steps to a logarithmic spiral.

 

I never had my own math this old before. I can’t remember what I was thinking when I wrote it.

[Genady]

On 5/9/2023 at 3:27 PM, Trurl said:

The authors wrote it working in just 1 week a year for 10 years.

Excellent read.

OK, I will read it, 1 week a year for 10 years.  

Seriously, thanks for the recommendation.

[Trurl]

@Genady

 

Did you read: Prime Numbers and the Riemann Hypothesis; Mazur and Stein?

I finished it Tuesday. Excellent read. It focused on the logical steps behind the hypothesis instead of deriving equations. There were equations but the book showed the reasoning behind them. I have never read a math work that gets you up to speed on such a complex problem so fast. 5 stars ⭐️

[Genady]

10 minutes ago, Trurl said:

@Genady

 

Did you read: Prime Numbers and the Riemann Hypothesis; Mazur and Stein?

I finished it Tuesday. Excellent read. It focused on the logical steps behind the hypothesis instead of deriving equations. There were equations but the book showed the reasoning behind them. I have never read a math work that gets you up to speed on such a complex problem so fast. 5 stars ⭐️

Not yet, but it is in the line. I have three other books to read before its turn comes.

I understand that it is a good book. I suspected so, that's why bought it to start with. I just don't see prime numbers having a very high priority in my education. Studying differential forms at the moment.

[Trurl]

Differential forms on a Riemann surface

Calculus applies to the properties of the graph. A differential equation. I only know the basics of such things. But if you get bored studying forms try analyzing the Zeta function 😜

Just looking at that link I posted I can’t even follow the notation let alone the math. But if differential forms have to deal with the addition and manipulation of functions then you would want to study the Riemann stuff.

[Genady]

53 minutes ago, Trurl said:

Differential forms on a Riemann surface

Calculus applies to the properties of the graph. A differential equation. I only know the basics of such things. But if you get bored studying forms try analyzing the Zeta function 😜

Just looking at that link I posted I can’t even follow the notation let alone the math. But if differential forms have to deal with the addition and manipulation of functions then you would want to study the Riemann stuff.

There are many kinds of Riemann stuff, pretty much everywhere in math. This specific stuff directly relates to Riemann geometry, and I don't know of a connection between this and prime numbers.