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

Deng Kaixin, Luo Senping

TL;DR

The paper advances the study of zeta and theta functions by establishing the signs of higher-order mixed derivatives, extending classical results on first derivatives. It combines modular invariance, 1D theta-function estimates, and the exponential theta expansion to prove positivity of a mixed derivative in a region and negativity of a higher-order mixed derivative on the modular domain, which in turn imply corresponding sign patterns for Epstein zeta derivatives. The authors then connect these derivative signs to lattice minimization problems, showing that extremal configurations within the fundamental domain can be detected via boundary analysis and standard zeta–theta relations. These results contribute to the mathematical understanding of lattice energetics and have potential implications for crystallization-type problems in two dimensions.

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.

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

TL;DR

The paper advances the study of zeta and theta functions by establishing the signs of higher-order mixed derivatives, extending classical results on first derivatives. It combines modular invariance, 1D theta-function estimates, and the exponential theta expansion to prove positivity of a mixed derivative in a region and negativity of a higher-order mixed derivative on the modular domain, which in turn imply corresponding sign patterns for Epstein zeta derivatives. The authors then connect these derivative signs to lattice minimization problems, showing that extremal configurations within the fundamental domain can be detected via boundary analysis and standard zeta–theta relations. These results contribute to the mathematical understanding of lattice energetics and have potential implications for crystallization-type problems in two dimensions.

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.
Paper Structure (3 sections, 13 theorems, 63 equations)

This paper contains 3 sections, 13 theorems, 63 equations.

Table of Contents

  1. Introduction and Statement of Main Results
  2. Estimates on 1-d theta functions
  3. Proof of Theorem \ref{['Th1']}

Key Result

Proposition 1.1

Signs of first order derivatives of $\zeta(s,z)$ and $\theta(\alpha,z)$ in fundamental domain.

Theorems & Definitions (20)

  • Definition 1: Apo1976Eva1973
  • Proposition 1.1: Cas1959Dia1964Enn1964aEnn1964bMon1988Ran1953
  • Theorem 1.1
  • Corollary 1.1
  • Lemma 2.1: Luo2023DL2024
  • Lemma 2.2
  • proof
  • Lemma 2.3: Luo2022Luo2023DL2024
  • proof
  • Lemma 2.4
  • ...and 10 more