I've counted the number of zeros of $\zeta$ by checking sign changes of $Z(t)$. I used a tolerance of 0.1 and found 649 zeros less than 1000, and then a tolerance of 0.05 to find 10,142 zeros less than 10,000. (if anyone can confirm these that would be great btw)
I know just "arbitrarily" picking a tolerance isn't really rigorous but I wanted to verify the Riemann Hypothesis first then go back and discuss why I chose each selection of $\delta$ likely using data from Odlyzko.
But to verify the RH I need to integrate $\zeta(s)$ over some closed rectangle in the complex plane with $0.4 < x < 0.6$ and $-1000 \leq y \leq 1000$. This should give me some complex number, which when divided by $2\pi i$ and combined with Cauchy's Residue theorem will give the number of singularities inside the rectangle. If this number agrees with the number of zeros found on the critical line then the RH has been verified up to the specified height.
So my question is: How do I evaluate this integral? Someone I was discussing this with mentioned that I could sum up the difference between each zero but surely the differences will just sum up to 1000 or 10,000 respectively?
Any help appreciated! Thanks in advance!