Is this number already known to be transcendental? Is there a survey about up-to-date trascendence results? - MathOverflow

Is this number already known to be transcendental? Is there a survey about up-to-date trascendence results?

2011-02-14T11:10:49Z

Is the number $\sum_{n=1}^\infty \frac{1}{2^{n^2}}$ known to be transcendental?

Is there a survey with up-to-date transcendence results?

Answer by Gerry Myerson for Is this number already known to be transcendental? Is there a survey about up-to-date trascendence results?

2011-02-14T11:27:38Z

There is a family of functions called theta-functions. One of them - I think the standard notation for it might be $\theta_3$ - is given by $\theta_3(z)=\sum z^{n^2}$, so (modulo any mistakes in the definition I've given) your number is $\theta_3(1/2)$. Now the theta-functions are very well-studied, and I suspect there is a lot of information out there about the transcendence of their values at rational arguments. So I've given you a keyword to aid your searches.

Answer by Andrey Rekalo for Is this number already known to be transcendental? Is there a survey about up-to-date trascendence results?

2011-02-14T11:37:59Z

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.