Finite Euler Products and the Riemann Hypothesis
by
Steve Gonek
University of Rochester

We begin by discussing approximations of the Riemann zeta-function by truncations of its

Dirichlet series and Euler product. We then construct a parameterized family of non-analytic

approximations to the zeta-function. Every function in the family satisfies a Riemann

hypothesis with the possible exception of a few zeros off the critical line. We show that 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.

Date received: June 15, 2008