# Spectral zeta functions

## TL;DR

This paper explores spectral zeta functions related to graphs, highlighting their structure, analogies with classical functions, and their appearance in various mathematical contexts, aiming to stimulate further research.

## Contribution

It provides a systematic study of spectral zeta functions associated with graphs, emphasizing their structure, analogies, and connections to diverse mathematical areas.

## Key findings

- Spectral zeta functions associated with graphs have a parallel structure to classical zeta functions.
- These functions appear in various contexts such as Eisenstein series, the Langlands program, and hypergeometric functions.
- Open problems are proposed to encourage further exploration in the field.

## Abstract

This paper discusses the simplest examples of spectral zeta functions, especially those associated with graphs, a subject which has not been much studied. The analogy and the similar structure of these functions, such as their parallel definition in terms of the heat kernel and their functional equations, are emphasized. Another theme is to point out various contexts in which these non-classical zeta functions appear. This includes Eisenstein series, the Langlands program, Verlinde formulas, Riemann hypotheses, Catalan numbers, Dedekind sums, and hypergeometric functions. Several open-ended problems are suggested with the hope of stimulating further research.