Request for the proof of a result from Ramanujan's letter to Hardy.

2012-07-02T17:01:58Z

Srinivasa Ramanujan in his first letter to G.H. Hardy stated many results for which he didn't give proofs. Among them the result taken from this link seems interesting :

If $$\int\limits_{0}^{\infty} \frac{\cos{nx}}{e^{2\pi\sqrt{x}}-1} \ dx = \phi(n)$$ then $\displaystyle\int\limits_{0}^{\infty} \frac{\sin{nx}}{e^{2\pi\sqrt{x}}-1} \ dx = \phi(n)-\frac{1}{2n} + \phi\biggl(\frac{\pi^2}{n}\biggr)\sqrt{\frac{2\pi^3}{n^3}}$.

The link also mentions that $\phi(n)$ is a complicated function. The following are certain special values and shows some values.

Questions which I would like to ask here are:

Where can I find the proof of the above result?
"The following are certain special values": Whats so special about the values?

Answer by Chandrasekhar for Request for the proof of a result from Ramanujan's letter to Hardy.

2012-07-03T06:53:03Z

Thanks "@Charles Matthews". I did email Prof. Berndt (after seeing your comment) and he suggested me to look at this paper:

The result appears here in Page 60. with proof.