A remark on Nesterenko’s theorem for Ramanujan functions

Abstract

We study the algebraic independence of values of the Ramanujan q-series $A_{2j+1}(q)=\sum_{n=1}^{\infty}n^{2j+1}q^{2n}/(1-q^{2n})$ or S _2j+1(q) (j≥0). It is proved that, for any distinct positive integers i, j satisfying $(i,j)\not=(1,3)$ and for any $q\in \overline{ \mathbb{Q}}$ with 0<|q|<1, the numbers A ₁(q), A _2i+1(q), A _2j+1(q) are algebraically independent over $\overline{ \mathbb{Q}}$ . Furthermore, the q-series A _2i+1(q) and A _2j+1(q) are algebraically dependent over $\overline{ \mathbb{Q}}(q)$ if and only if (i,j)=(1,3).