Is the number $\sum_{n=1}^\infty \frac{1}{2^{n^2}}$ known to be transcendental?
Is there a survey with uptodate transcendence results?
Is the number $\sum_{n=1}^\infty \frac{1}{2^{n^2}}$ known to be transcendental? Is there a survey with uptodate transcendence results? 


I have checked with Introduction to Algebraic Independence Theory, where it is mentioned in the preface (p. V) that
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. 339350, 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. 202203. 


There is a family of functions called thetafunctions. 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 thetafunctions are very wellstudied, 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. 

