Algebraic results for the values 𝜗3(mτ) and 𝜗3(nτ) of the Jacobi theta-constant

Let $ϑ_{3} (τ) = 1 + 2 \sum_{ν = 1}^{\infty} e^{π i ν^{2} τ}$ denote the classical Jacobi theta-constant. We prove that the two values $ϑ_{3} (m τ)$ and $ϑ_{3} (n τ)$ are algebraically independent over $ℚ$ for any $τ$ in the upper half-plane such that $q = e^{π i τ}$ is an algebraic number, where $m, n \geq 2$ are distinct integers.

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Keywords

algebraic independence, Jacobi theta-constants, modular functions

Mathematical Subject Classification 2010

Primary: 11J85

Secondary: 11J91, 11F27

Milestones

Received: 10 January 2018

Revised: 18 June 2018

Accepted: 3 July 2018

Published: 11 August 2018

Authors

Carsten Elsner
Fachhochschule für die Wirtschaft University of Applied Sciences Hannover Germany

Florian Luca
School of Mathematics University of the Witwatersrand Johannesburg South Africa	Max Planck Mathematical Institute Bonn Germany	Department of Mathematics Faculty of Sciences University of Ostrava Ostrava Czech Republic

Yohei Tachiya
Graduate School of Science and Technology Hirosaki University Hirosaki Japan

Vol. 8, No. 1, 2019

Carsten Elsner, Florian Luca and Yohei Tachiya

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification 2010

Milestones

Authors