Functiones et Approximatio Commentarii Mathematici

Algebraic independence results for values of theta-constants

Carsten Elsner

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Abstract

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.

Article information

Source
Funct. Approx. Comment. Math., Volume 52, Number 1 (2015), 7-27.

Dates
First available in Project Euclid: 20 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.facm/1426857031

Digital Object Identifier
doi:10.7169/facm/2015.52.1.1

Mathematical Reviews number (MathSciNet)
MR3326120

Zentralblatt MATH identifier
1325.11074

Subjects
Primary: 11J85: Algebraic independence; Gelʹfond's method
Secondary: 11J91: Transcendence theory of other special functions 11F27: Theta series; Weil representation; theta correspondences

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

Citation

Elsner, Carsten. Algebraic independence results for values of theta-constants. Funct. Approx. Comment. Math. 52 (2015), no. 1, 7--27. doi:10.7169/facm/2015.52.1.1. https://projecteuclid.org/euclid.facm/1426857031


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