Finite Euler products and the Riemann Hypothesis
Title: Finite Euler products and the Riemann Hypothesis
Speaker: Steve M. Gonek (University of Rochester)
Time: 20:00-21:00 Beijing time, Jan 10, 2023
Venue: BIMSA 1118
Zoom ID: 293 812 9202 Passcode: BIMSA
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.