What are the different methods using contour integration that can be used to count the non-trivial zeros of the zeta function?
I know of Backlund's exact formula which is similar to Riemann-von Mangoldt formula. What are the other ways?
If there aren't many other different methods, then where can I find the exact derivation, like from scratch to end of Riemann-von Mangoldt formula? I have managed half of it till the point I can derive Backlund's exact formula. Also an evaluation of the limitations would be nice.
I know Titchmarsh's works are useful but what other resources can help me?