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

2025/01/02 by Deng Kaixin, Kaixin, Deng, Luo Senping +1
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.2501.01265

openalex publication_date 2025/01/02 · openalex created_date 2025/01/04 · openalex updated_date 2026/07/28

Abstract

In the 1950s, 1960s and 1988, number theorists Rankin \citeRan1953, Cassels \citeCas1959, Ennola \citeEnn1964a, Diananda \citeDia1964, and Montgomery \citeMon1988 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.

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