An explicit prime geodesic theorem for discrete tori and the hypergeometric functions

TL;DR

This paper derives an explicit, non-asymptotic prime geodesic theorem for discrete tori using advanced hypergeometric functions, expanding the understanding of spectral properties of these graph structures.

Contribution

It introduces a novel explicit prime geodesic theorem for discrete tori, utilizing Lauricella hypergeometric functions to generalize classical polynomial methods.

Findings

Explicit prime geodesic formula for discrete tori

Connection between hypergeometric functions and graph spectra

Generalization of Jacobi polynomials via Lauricella functions

Abstract

The discrete tori are graph analogues of the real tori, which are defined by the Cayley graphs of a finite product of finite cyclic groups. In this paper, using the theory of the heat kernel on the discrete tori established by Chinta, Jorgenson and Karlsson, we derive an explicit prime geodesic theorem for the discrete tori, which is not an asymptotic formula. To describe the formula, we need generalizations of the classical Jacobi polynomials, which are defined by the Lauricella multivariable hypergeometric function of type C.