Signs of high order derivatives for the theta and Epstein zeta functions and application

TL;DR

This paper investigates the signs of higher order derivatives of Epstein zeta and theta functions, extending previous work on first derivatives, and applies these findings to lattice minimization problems.

Contribution

It introduces new results on the signs of higher order derivatives of Epstein zeta and theta functions, expanding the understanding of their properties.

Findings

Signs of higher order derivatives derived

Applications to lattice minimization demonstrated

Extends classical results from first derivatives to higher orders

Abstract

In the 1950s, 1960s and 1988, number theorists Rankin \cite{Ran1953}, Cassels \cite{Cas1959}, Ennola \cite{Enn1964a}, Diananda \cite{Dia1964}, and Montgomery \cite{Mon1988} derived the signs of first order derivatives of Epstein zeta and theta functions, respectively. In this note, we shall derive the signs of higher order derivatives of such functions. Application to lattice minimization problems will be given.