I've just been reading through my number theory lecture notes and I noticed this line of reasoning:(adsbygoogle = window.adsbygoogle || []).push({});

***Assuming the Rieman hypothesis***

[tex]\sum_{\gamma > 0}\sin(\gamma \log x) \frac{-(1/2)^2}{\gamma((1/2)^2 + \gamma^2)} = O(1)[/tex]

by comparisson with [tex]\sum_{n \ge 1}\frac{1}{n^2}[/tex].

Let me explain the context a little. Importantly the gamma are the non-trivial zeros of the zeta function in the upper half plane. (The goal is to obtain an expression which demonstrates a connection between the Von-Mangolt explicit formula and Fourier series if anyone's interested but that is really not important for my question.)

Now as far as I can see the statement is true by considering absolute convergence etc. but surely this requires also that the spacing of the zeros on the line Re(z) = 1/2 is sufficenty sparse so that the comparison is valid.

There is no mention anywhere else in the course of this being a fact. Is it true?

How much do we know about properties of the distribution? I suppose the above question concerns the existence of a lower bound on the spacing.

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# Spacing of Riemann's Zeros

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