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?
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$\begingroup$ Would you please tell us what is your motivations?Thanks. $\endgroup$– awllowerCommented Feb 14, 2011 at 11:14
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$\begingroup$ I'm writing a paper in which I show that a variation of the number above is an l^2-Betti number of something (cf. arxiv.org/abs/1004.2030) $\endgroup$– Łukasz GrabowskiCommented Feb 14, 2011 at 12:44
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$\begingroup$ See also this answer of mine for a proof that this and other similar numbers are transcendental, and some related links. mathoverflow.net/questions/41609/… $\endgroup$– George LowtherCommented Feb 14, 2011 at 15:18
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$\begingroup$ Sorry, made a mistake there. This series doesn't converge nearly fast enough for the results in the other question to apply. $\endgroup$– George LowtherCommented Feb 14, 2011 at 16:11
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1$\begingroup$ @George: I think you confuse the number from my question with $\sum \frac{1}{2^{2^n}}$. $\endgroup$– Łukasz GrabowskiCommented Feb 14, 2011 at 16:18
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2 Answers
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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.
answered Feb 14, 2011 at 11:37
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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.
answered Feb 14, 2011 at 11:27