On the Instability of the Riemann Hypothesis over Finite Fields

2010/08/03 by Paul M. Gauthier, Gauthier, P. M., Николай Тарханов +1
Mathematics · #11M99 #30E10 #30K99 #Analytic Number Theory Research #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.1008.0499

openalex publication_date 2010/08/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• The Riemann Hypothesis for Function Fields over a Finite Field
• On the irreducibility of the two variable zeta-function for curves over finite fields
• Asymptotic properties of zeta functions over finite fields
• Generalised root identities for zeta functions of curves over finite fields
• Riemann Hypothesis for Non-Abelian Zeta Functions of Curves over Finite Fields