I wont pretent to fully understand the hpothesis itself because of lack of knowledge of complex numbers (though i have basic understanding of imaginary numbers i dont understand the "critical line" or why infinite 0's sit on it).

 

However i grasp the concept of the zeta function and how generally exponents create a convergence.

 

I specifically want to know how using the pattern 1-(1/2^s) + 1-(1/3^s) + 1-(1/5^s) + 1-(1/7^s) etc relates to the zeta function. The example i have doesnt clarify how the zeta function and this prime pattern relate exactly so any information would be gladly accepted. (Also any examples of complex numbers that sit on this critical line).

 

Regards.

so, let's start from the beginning.

the harmonic series:

1/1 +1/2 +1/3 +1/4 ... grows indefinitely.

however the series 1/1 +1/2^2 +1/3^2 +1/4^2 etc. aproaches a value, specifically pi/6 (or was it pi^2/6? well the point is its finite)

in general 1/2^s +1/3^s +1/4^s ... aproaches an interesting value (something in terms of pi.) whenever s is even. now if we modify the equation in a few ways: first well do 1- each term.

1- 1/2^s +1-1/3^s +1-1/4^s...

then we'll take the reciprical of each term.

1/(1-1/2^s) +1/(1-1/3^s) +1/(1-1/4^s)...

then we'll multiply rather than add.

1/(1-1/2^s)*1/(1-1/3^s)*1/(1-1/4^s)...

and finnally we'l exclude any non prime power terms.

1/(1-1/2^s)*1/(1-1/3^s)*1/(1-1/5^s)*1/(1-1/7^s)...

it turns out this is exaclty equvalent to our original equation 1/1 +1/2^s +1/3^s... (its possible to prove this but admittedly i don't know how, im not expert.

although this looks ugly, this multiplication means that if we can get the right side to be non-trivally zero we have information we can use to estimate the gap between primes.

all trival zero occur whenever s is even. all non-trivail zeros occur when s = a +1/2*sqrt(-1) where a is some unkown value. or at least this is the hypothesis, its never been proven.

Edited October 5, 201510 yr by phillip1882