An integral - Riemann's derivation of number of primes
OK so I am not sure if this problem is analysis, number theory or calculus.
I am reading an English translation of Riemann's 1859 paper on the number of primes and cannot follow this bit.
He says consider this integral:
From infinity in a positive sense around a domain that includes 0 but does not include any other point of discontinuity. This is then equal to:
I would like to fill in the details of this step but have been unsuccessful. I have tried:
Finding the residue at z=0 by solving:
I also tried generalising x to z and integrating around the path from inf, to to to to
This gave interesting results when either; tended to 0 equals pi or equals pi/2.
But non-the less I have not been able to get the required result. Is one of my approaches right? Is there a better approach?