I'm trying to locate the first 1000 zeros of $\zeta(s)$ and not sure about the best way to go about it. I was considering the Newton-Raphson method but can'd find a good way to code $\zeta'(s)$ in python as I can't find a functional equation anywhere.
I'm using a simple 'while' loop to locate sign changes of $Z(t)$ to count zeros (would also appreciate if anyone knows a quicker way to count the zeros of $\zeta(s)$ btw) and have found 649 zeros below 1000 and 10,142 zeros below 10,000.
I now need to locate these zeros so I can sum the arguments of each complex zero. I'm led to believe this will give me $$\int_{\gamma} \zeta(s) ~ ds$$ where $\gamma$ is the contour of length 1000 or 10,000 respectively. ie: $$\int_{\gamma} \zeta(s) ~ ds = 2\pi \cdot \sum_{s' \in S} \arg{(s')}$$ where $S$ is the set of all zeros below 1000 or 10,000 around the critical strip. I will first calculate this then back it up rigorously.
Thus if I am able to locate all zeros and sum their arguments, then divide this number by $2 \pi$ I will be able to directly compare this result with the number of zeros found earlier, the goal being to verify the Riemann Hypothesis up to a certain height.
Any help very much appreciated!
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