Special values of spectral zeta functions of graphs and Dirichlet L-functions

TL;DR

This paper explores the deep connections between special values of Dirichlet L-functions and spectral zeta functions of graphs, revealing they determine each other through a combinatorial derivative formula derived from spectral analysis.

Contribution

It establishes a natural relationship between Dirichlet L-functions and spectral zeta functions of cycle graphs, introducing a novel combinatorial derivative formula.

Findings

Special values of Dirichlet L-functions and spectral zeta functions determine each other.

A combinatorial derivative formula links spectral zeta functions to L-functions.

The results bridge number theory and spectral graph theory.

Abstract

In this paper, we establish relations between special values of Dirichlet $L$-functions and that of spectral zeta functions or $L$-functions of cycle graphs. In fact, they determine each other in a natural way. These two kinds of special values were bridged together by a combinatorial derivative formula obtained from studying spectral zeta functions of the first order self-adjoint differential operators on the unit circle.