Finite Euler products and the Riemann Hypothesis

TL;DR

This paper explores the relationship between the Riemann Hypothesis and the approximation of the zeta-function by Euler products, introduces a family of functions mimicking zeta zeros, and discusses implications for zero distribution and related L-functions.

Contribution

It establishes a connection between the Riemann Hypothesis and Euler product approximations, constructs a new family of functions with similar zero properties, and provides insights into zero distribution of L-functions.

Findings

If RH is true, the zeta-function is well approximated by short Euler products in a critical region.

The constructed functions have zeros mostly on the critical line, with simple and repelling zeros.

Zeros of these functions closely match those of the zeta-function and converge as parameters vary.

Abstract

We show that if the Riemann Hypothesis is true, then in a region containing most of the right-half of the critical strip, the Riemann zeta-function is well approximated by short truncations of its Euler product. Conversely, if the approximation by products is good in this region, the zeta-function has at most finitely many zeros in it. We then construct a parameterized family of non-analytic functions with this same property. With the possible exception of a finite number of zeros off the critical line, every function in the family satisfies a Riemann Hypothesis. Moreover, when the parameter is not too large, they have about the same number of zeros as the zeta-function, their zeros are all simple, and they "repel". The structure of these functions makes the reason for the simplicity and repulsion of their zeros apparent and suggests a mechanism that might be responsible for the corresponding properties of the zeta-function's zeros. Computer evidence suggests that the zeros of functions in the family are remarkably close to those of the zeta-function (even for small values of the parameter), and we show that they indeed converge to them as the parameter increases. Furthermore, between zeros of the zeta-function, the moduli of functions in the family tend to twice the modulus of the zeta-function. Both assertions assume the Riemann Hypothesis. We end by discussing analogues for other L-functions and show how they give insight into the study of the distribution of zeros of linear combinations of L-functions.