On ratios of theta functions

TL;DR

This paper classifies extremal ratios of theta and Epstein zeta functions, highlighting the significance of the hexagonal lattice, with applications in conformal field theory and crystallization mathematics.

Contribution

It provides a complete classification of minimizers and maximizers for ratios of theta and Epstein zeta functions, emphasizing the role of the hexagonal lattice.

Findings

Hexagonal lattice is pivotal in extremal ratios.

Complete classification of minimizers and maximizers.

Results apply to conformal field theory and crystallization.

Abstract

Motivated by the average partition function of c free bosons $($Afhkami-Jeddi et al. \cite{Afhk2021}$)$ and the average of the genus 1 partition function over the Narain moduli space $($Maloney-Witten \cite{Witten2020}$)$, we investigate ratios of theta functions. In this paper, we completely classify the minimizers (or maximizers) for ratios of theta and Epstein zeta functions. We find that the hexagonal lattice plays a pivotal role there. These results have direct applications in conformal and Liouville field theory via partition functions. Additionally, they yield the minima of differences of theta and Epstein zeta functions, which have implications for the mathematics of crystallization and interacting particle theory (\cite{Bet2016,Bet2019AMP}).