Riemann Zeta HypothesisDate: 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/ |
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