A functional relation for L-functions of graphs equivalent to the Riemann Hypothesis for Dirichlet L-functions

TL;DR

This paper introduces L-functions for finite graphs, especially cycles, and explores their connection to the Riemann Hypothesis, suggesting an equivalence between graph-based L-functions' properties and classical number theory conjectures.

Contribution

It establishes a novel link between graph L-functions and the Riemann Hypothesis, proposing a new perspective on the classical conjecture through graph theory.

Findings

Asymptotic functional equation for graph L-functions relates to the Generalized Riemann Hypothesis.

Positivity of graph L-functions correlates with zeros of Dirichlet L-functions.

Potential equivalence between graph L-functions' behavior and classical number theory conjectures.

Abstract

In this note we define L-functions of finite graphs and study the particular case of finite cycles in the spirit of a previous paper that studied spectral zeta functions of graphs. The main result is a suggestive equivalence between an asymptotic functional equation for these L-functions and the corresponding case of the Generalized Riemann Hypothesis. We also establish a relation between the positivity of such functions and the existence of real zeros in the critical strip of the classical Dirichlet L-functions with the same character.