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edited Feb 14 2011 at 12:04

I have checked with Introduction to Algebraic Independence Theory, where it is mentioned in the preface (p. V) that

D. Bertrand and independently D. Duverney, Ke. Nishioka, Ku. Nishioka, I. Shiokawa (DNNS) deduced results on algebraic independence of the values of theta-functions at algebraic points and in particular derived the transcendence of the sums $\sum_{n=1}^\infty q^{n^2}$ for any algebraic $q$ satisfying $0 < |q| < 1$.

The precise references are not given but a little googling turned up the paper by D. Bertrand, "Theta Functions and Transcendence", The Ramanujan Journal, Vol. 1 (1997), pp. 339-350, which seems to be relevant. The second reference is DNNS, "Transcendence of Jacobi's theta series", Proc. Japan Acad. Ser. A Math. Sci., Vol. 72 (1996), pp. 202-203.

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edited Feb 14 2011 at 11:52

Andrey Rekalo
14k●1●34●69

I have checked with Introduction to Algebraic Independence Theory, where it is mentioned in the preface (p. V) that

D. Bertrand and independently D. Duverney, Ke. Nishioka, Ku. Nishioka, I. Shiokawa (DNNS) deduced results on algebraic independence of the values of theta-functions at algebraic points and in particular derived the transcendence of the sums $\sum_{n=1}^\infty q^{n^2}$ for any algebraic $q$ satisfying $0 < |q| < 1$.

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answered Feb 14 2011 at 11:37

Andrey Rekalo
14k●1●34●69

I have checked with Introduction to Algebraic Independence Theory, where it is mentioned in the preface (p. V) that

D. Bertrand and independently D. Duverney, Ke. Nishioka, Ku. Nishioka, I. Shiokawa (DNNS) deduced results on algebraic independence of the values of theta-functions at algebraic points and in particular derived the transcendence of the sums $\sum_{n=1}^\infty q^{n^2}$ for any algebraic $q$ satisfying $0 < |q| < 1$.