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2019 Algebraic results for the values $\vartheta_3(m\tau)$ and $\vartheta_3(n\tau)$ of the Jacobi theta-constant
Carsten Elsner, Florian Luca, Yohei Tachiya
Mosc. J. Comb. Number Theory 8(1): 71-79 (2019). DOI: 10.2140/moscow.2019.8.71

Abstract

Let ϑ 3 ( τ ) = 1 + 2 ν = 1 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 for any τ in the upper half-plane such that q = e π i τ is an algebraic number, where m , n 2 are distinct integers.

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Carsten Elsner. Florian Luca. Yohei Tachiya. "Algebraic results for the values $\vartheta_3(m\tau)$ and $\vartheta_3(n\tau)$ of the Jacobi theta-constant." Mosc. J. Comb. Number Theory 8 (1) 71 - 79, 2019. https://doi.org/10.2140/moscow.2019.8.71

Information

Received: 10 January 2018; Revised: 18 June 2018; Accepted: 3 July 2018; Published: 2019
First available in Project Euclid: 3 December 2018

zbMATH: 07063264
MathSciNet: MR3864309
Digital Object Identifier: 10.2140/moscow.2019.8.71

Subjects:
Primary: 11J85
Secondary: 11F27 , 11J91

Keywords: algebraic independence , Jacobi theta-constants , modular functions

Rights: Copyright © 2019 Mathematical Sciences Publishers

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