This integral is called Vardi's Integral (presumably named after someone named Vardi ).
The answer to this may deserve a technical paper (actually you can find it in literature) and may require considerable knowledge in number theory. Here is another one,
Last edited by mr fantastic; Sep 18th 2009 at 08:49 AM.
Reason: Restored original reply
From here you could use contour integration. But how about integrating under the integral sign?
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The remaining is no easy task. It uses the Dirichlet’s L-functions. There are many beautiful definite integrals with beautiful results out there but the solution or proof procedures are just painstaking.
Last edited by mr fantastic; Sep 18th 2009 at 08:49 AM.
Reason: Restored orignal reply