Finite Euler products and the Riemann Hypothesis

Abstract

We investigate approximations of the Riemann zeta function by truncations of its Dirichlet series and Euler product, and then construct a parameterized family of non-analytic approximations to the zeta function. Apart from a few possible exceptions near the real axis, each function in the family satisfies a Riemann Hypothesis. When the parameter is not too large, the functions have roughly the same number of zeros as the zeta function, their zeros are all simple, and they repel. In fact, if the Riemann hypothesis is true, the zeros of these functions converge to those of the zeta function as the parameter increases, and between zeros of the zeta function the functions in the family tend to twice the zeta function. They may therefore be regarded as models of the Riemann zeta function. The structure of the functions explains the simplicity and repulsion of their zeros when the parameter is small. One might therefore hope to gain insight from them into the mechanism responsible for the corresponding properties of the zeros of the zeta function.

Speaker

Prof. Gonek’s research interests are in the field of analytic number theory, particularly multiplicative number theory, the theory of the Riemann zeta-function, L-functions, and the distribution of prime numbers. Some of his work has focused on moments of the Riemann zeta-function, discrete mean value theorems for the zeta-function and L-functions, and the development and application of random matrix models for the zeta-function. One goal of this work is to better understand the behavior (the distribution of zeros, maximal order, and so on) of the zeta and L-functions themselves. Another is to determine connections between these behaviors and various arithmetical problems. Prof. Gonek has also worked on questions relating to the distribution of multiplicative inverses and primitive roots in residue classes modulo a prime.