Open Access
March 2015 Algebraic independence results for values of theta-constants
Carsten Elsner
Funct. Approx. Comment. Math. 52(1): 7-27 (March 2015). DOI: 10.7169/facm/2015.52.1.1


Let $\theta(q)=1+2\sum_{\nu=1}^{\infty} q^{\nu^2}$ denote the Thetanullwert of the Jacobi Zeta function \[\theta(z|\tau) =\sum_{\nu=-\infty}^{\infty} e^{\pi i\nu^2\tau + 2\pi i\nu z}.\] For algebraic numbers $q$ with $0<|q|<1$ we prove the algebraic independence over $\mathbb{Q}$ of the numbers $\theta(q^n)$ and $\theta(q)$ for $n=2,3,\dots,12$ and furthermore for all $n\geq 16$ which are powers of two. An application for $n=5$ proves the transcendence of the number \[\sum_{j=1}^{\infty} {(-1)}^j \Big( \frac{j}{5}\Big) \frac{jq^j}{1-q^j}.\] Similar results are obtained for numbers related to modular equations of degree 3, 5, and 7.


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Carsten Elsner. "Algebraic independence results for values of theta-constants." Funct. Approx. Comment. Math. 52 (1) 7 - 27, March 2015.


Published: March 2015
First available in Project Euclid: 20 March 2015

zbMATH: 1325.11074
MathSciNet: MR3326120
Digital Object Identifier: 10.7169/facm/2015.52.1.1

Primary: 11J85
Secondary: 11F27 , 11J91

Keywords: algebraic independence , independence criterion , modular equations , Nesterenko's theorem , theta-constants

Rights: Copyright © 2015 Adam Mickiewicz University

Vol.52 • No. 1 • March 2015
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