Previous issue ·  Next issue ·  Recently posted articles ·  Most recent issue · All issues   
Home Overview Authors Editorial Contact Subscribe

Journal of the Ramanujan Mathematical Society

Volume 35, Issue 1, March 2020  pp. 71–80.

Algebraic independence results for the values of the theta-constants and some identities

Authors:  Carsten Elsner, Masanobu Kaneko and Yohei Tachiya
Author institution:Fachhochschule für die Wirtschaft, University of Applied Sciences, Freundallee 15, D-30173 Hannover, Germany

Summary:  In the present work, we give algebraic independence results for the values of the classical theta-constants ϑ 2 (τ), ϑ 3(τ), and ϑ 4(τ). For example, the two values ϑ α (mτ) and ϑ β (nτ) are algebraically independent over Q for any τ in the upper half-plane when e{π iτ} is an algebraic number, where m,n≥ 1 are integers and α, β ∈ {2,3,4} with (m,α) ≠ (n,β). This algebraic independence result provides new examples of transcendental numbers through some identities found by S.~Ramanujan. We additionally give some explicit identities among the three theta-constants in particular cases.

Contents   Full-Text PDF