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How can we prove what are, say the first 4 non-trivial zeroes of the Riemann $\zeta$ on the critical line $Re(z_j)=\frac{1}{2}$, $j=1,2,3,4$ the first two with negative imaginary part and the second two with positive imaginary part? What can in general be said about the frequency of the imaginary part with which all the non-trivial zeroes occur, if something at all?

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As for computing the zeros, there's a good discussion in "Riemann's Zeta Function," by H. M. Edwards. I don't understand what you mean by "the frequency of the imaginary part." No imaginary part can occur more than once (unless the hypothesis is false.)

Here is a student paper on the subject that probably owes a lot to Edwards's discussion.

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  • $\begingroup$ Can you please give me a link to that discussion? I meant something in this style: the largest gap between two imaginary parts is less then 10 or some easy function depending on $n$ in the $n-$th non-trivial zero. $\endgroup$ – user122424 Oct 19 '18 at 15:23
  • $\begingroup$ I don't have a link. It's a book. $\endgroup$ – saulspatz Oct 19 '18 at 15:25

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