Request for the proof of a result from Ramanujan's letter to Hardy. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T16:52:48Z http://mathoverflow.net/feeds/question/101159 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/101159/request-for-the-proof-of-a-result-from-ramanujans-letter-to-hardy Request for the proof of a result from Ramanujan's letter to Hardy. Chandrasekhar 2012-07-02T17:01:58Z 2012-07-04T10:55:49Z <p><a href="http://en.wikipedia.org/wiki/Srinivasa_Ramanujan" rel="nofollow">Srinivasa Ramanujan</a> in his <a href="http://books.google.co.in/books?id=Of5G0r6DQiEC&amp;lpg=PP1&amp;pg=PA21#v=onepage&amp;q&amp;f=false" rel="nofollow">first letter</a> to <a href="http://en.wikipedia.org/wiki/G.H._Hardy" rel="nofollow">G.H. Hardy</a> stated many results for which he didn't give proofs. Among them the result taken from <a href="http://books.google.co.in/books?id=Of5G0r6DQiEC&amp;lpg=PP1&amp;pg=PA24#v=onepage&amp;q&amp;f=false" rel="nofollow">this link</a> seems interesting :</p> <ul> <li>If $$\int\limits_{0}^{\infty} \frac{\cos{nx}}{e^{2\pi\sqrt{x}}-1} \ dx = \phi(n)$$ then $\displaystyle\int\limits_{0}^{\infty} \frac{\sin{nx}}{e^{2\pi\sqrt{x}}-1} \ dx = \phi(n)-\frac{1}{2n} + \phi\biggl(\frac{\pi^2}{n}\biggr)\sqrt{\frac{2\pi^3}{n^3}}$. </li> </ul> <p>The link also mentions that $\phi(n)$ is a complicated function. The following are certain special values and shows some values.</p> <p>Questions which I would like to ask here are:</p> <ul> <li><p>Where can I find the proof of the above result?</p></li> <li><p>"<em>The following are certain special values</em>": Whats so special about the values?</p></li> </ul> http://mathoverflow.net/questions/101159/request-for-the-proof-of-a-result-from-ramanujans-letter-to-hardy/101213#101213 Answer by Chandrasekhar for Request for the proof of a result from Ramanujan's letter to Hardy. Chandrasekhar 2012-07-03T06:53:03Z 2012-07-04T10:55:49Z <p>Thanks "@Charles Matthews". I did email Prof. Berndt (after seeing your comment) and he suggested me to look at this paper:</p> <ul> <li><em><a href="http://books.google.co.in/books?id=oSioAM4wORMC&amp;lpg=PA357&amp;dq=Ramanujan%20collected%20papers&amp;pg=PA59#v=onepage&amp;q=Ramanujan%20collected%20papers&amp;f=false" rel="nofollow">Some definite integrals connected with Gauss's sums</a></em>, Mess. Math. 44 (1915), 75-85.</li> </ul> <p>The result appears here in <a href="http://books.google.co.in/books?id=oSioAM4wORMC&amp;lpg=PA357&amp;dq=Ramanujan%2520collected%2520papers&amp;pg=PA60#v=onepage&amp;q=Ramanujan%2520collected%2520papers&amp;f=false" rel="nofollow">Page 60.</a> with proof.</p>