Riemann Zeta Hypothesis

Date: 09/28/97 at 14:18:50
From: Selim Tezel
Subject: Riemann Zeta Hypothesis

Dear Dr. Math,

Thanks for helping out and for your time. I have been puzzled by the 
zeta function and its relation to the prime number theory. I have not 
been able to find good resources to help me understand. I have been 
especially puzzled by the claim that -2, -4, -5, ... are "trivial" 
zeros of the function. How so? It does not seem to me to be trivial at 
all. If the hypothesis of Riemann about the zeta function is proven, 
why will this have a great effect on prime number theory?  

Many thanks for your time.  Best wishes,

Selim Tezel
Math instructor at Robert College High School of Istanbul Turkey

Date: 09/29/97 at 15:27:35
From: Doctor Wilkinson
Subject: Re: Riemann Zeta Hypothesis

If you can find a copy of Titchmarsh's book on the Riemann zeta 
function, it explains all of this. But briefly, the reason the 
negative even integers are called trivial zeros is that it is 
relatively easy to prove that they are zeros. It is much harder to 
prove anything about the complex zeroes. Triviality is a matter of 
degree! But the general idea is that there is an equation relating 
zeta(s) and zeta(1-s) (the "functional equation of the zeta 
function").  The equation has the form

 gamma(s/2) * zeta(s) * (some power of pi) =
 gamma((1-s)/2) * zeta (1-s) * (some other power of pi).

Now it is easy to show that the gamma function has poles at the 
negative integers, and from this and the functional equation you can 
show that the zeta function has zeros at the negative even integers.

The connection between the zeta function and prime numbers is given by
Euler's product formula. It turns out that you can get estimates for 
the distribution of prime numbers in the form of contour integrals 
with zeta(s) in the denominator. To improve the estimates you want to 
move parts of the contour to the left as far as possible. But you 
can't move the contour across zeros of the function. If you know there 
are no zeros to the right of the line Re s = 1/2, you can improve the 
estimates.

By the way, I taught at Bogazici University for two very happy years
(1976-1978).  I have many fond memories of Turkey.

-Doctor Wilkinson,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/