On the Instability of the Riemann Hypothesis over Finite Fields

TL;DR

This paper demonstrates the ability to approximate zeta-functions of curves over finite fields with meromorphic functions that either satisfy or violate the Riemann hypothesis, exploring their value distribution through Nevanlinna theory.

Contribution

It introduces methods to approximate finite field zeta-functions with functions that do or do not satisfy the Riemann hypothesis, and analyzes their value distribution.

Findings

Approximation of zeta-functions by meromorphic functions with controlled properties

Existence of functions approximating the Riemann hypothesis failure

Application of Nevanlinna theory to zeta-functions over finite fields

Abstract

We show that it is possible to approximate the zeta-function of a curve over a finite field by meromorphic functions which satisfy the same functional equation and moreover satisfy (respectively do not satisfy) the analogue of the Riemann hypothesis. In the other direction, it is possible to approximate holomorphic functions by simple manipulations of such a zeta-function. We also consider the value distribution of zeta-functions of function fields over finite fields from the viewpoint of Nevanlinna theory.