Is this number already known to be transcendental? Is there a survey about up-to-date trascendence results? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T11:14:30Z http://mathoverflow.net/feeds/question/55397 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/55397/is-this-number-already-known-to-be-transcendental-is-there-a-survey-about-up-to Is this number already known to be transcendental? Is there a survey about up-to-date trascendence results? Łukasz Grabowski 2011-02-14T11:10:49Z 2011-02-14T12:04:34Z <p>Is the number $\sum_{n=1}^\infty \frac{1}{2^{n^2}}$ known to be transcendental?</p> <p>Is there a survey with up-to-date transcendence results?</p> http://mathoverflow.net/questions/55397/is-this-number-already-known-to-be-transcendental-is-there-a-survey-about-up-to/55398#55398 Answer by Gerry Myerson for Is this number already known to be transcendental? Is there a survey about up-to-date trascendence results? Gerry Myerson 2011-02-14T11:27:38Z 2011-02-14T11:27:38Z <p>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. </p> http://mathoverflow.net/questions/55397/is-this-number-already-known-to-be-transcendental-is-there-a-survey-about-up-to/55400#55400 Answer by Andrey Rekalo for Is this number already known to be transcendental? Is there a survey about up-to-date trascendence results? Andrey Rekalo 2011-02-14T11:37:59Z 2011-02-14T12:04:34Z <p>I have checked with <a href="http://books.google.com.ua/books?id=liYae-vUZs4C&amp;printsec=frontcover&amp;dq=introduction+to+algebraic+independence+theory&amp;source=bl&amp;ots=6Kf4QYSDlS&amp;sig=bJ2TfJ5Fi54ktqf56svrBYIkXyk&amp;hl=en&amp;ei=2BNZTfDtAcWSOq_PpPoE&amp;sa=X&amp;oi=book_result&amp;ct=result&amp;resnum=1&amp;ved=0CBcQ6AEwAA#v=onepage&amp;q&amp;f=false" rel="nofollow"><em>Introduction to Algebraic Independence Theory</em></a>, where it is mentioned in the preface (p. V) that </p> <blockquote> <p>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 $0 &lt; |q| &lt; 1$. </p> </blockquote> <p>The precise references are not given but a little googling turned up the paper by D. Bertrand, <a href="http://www.springerlink.com/content/u11786l337374231/" rel="nofollow">"Theta Functions and Transcendence"</a>, <em>The Ramanujan Journal</em>, Vol. 1 (1997), pp. 339-350, which seems to be relevant. The second reference is DNNS, <a href="http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&amp;id=pdf_1&amp;handle=euclid.pja/1195510210" rel="nofollow">"Transcendence of Jacobi's theta series"</a>, <em>Proc. Japan Acad. Ser. A Math. Sci.</em>, Vol. 72 (1996), pp. 202-203.</p>