As part of my MSc I am reviewing a paper. The paper is a review on the statistical distribution of the unfolded zeros (see below) of the Reimann functional equation. In the paper there is a sentence:

The mean density of the non-zero-trivial zeros increases logarithmically with height $t$ up the critical line. Specifically, defining unfolded zeros by $$ w_n = t_n \frac{1}{2\pi}\log{\frac{t_n}{2\pi}} $$ it is known that $$ \lim_{W\rightarrow \infty}\frac{1}{W}\#\{w_n < W\} = 1 $$

What does the bold above mean? More specifically, how does he know that?

Here it is assumed that the Riemann hypothesis is true. That is, $\zeta{(1/2 + it)} = 0$ has non-trivial solutions only when $t=t_n \in \mathbb{R}$.

I am not sure if I am clear enough, please specify for clarity if I am not.

Please note that I am just about to start my first module in analytic number theory, so please pitch any help accordingly.


The first sentence of the quotation should read "The mean density of the non-trivial zeros increases logarithmically with height $t$ up the critical line". What it means is defined exactly in the second sentence. Intuitively this means that, if we were to "stretch" the critical line, with the overall degree of stretching increasing logarithmically as we proceed up the line, we would separate the zeros so that the average distance between them remains constant along the line. Without this stretching, the zeros tend to cluster more tightly as we go along the line, with the overall clustering density increasing logarithmically. The continuous form of the stretching factor can be stated as$$\dfrac{1}{2\pi}\ln\dfrac{t}{2\pi}$$at height $t$.

  • $\begingroup$ Thank you. So does the use of the log come from the prime number theorem? $\endgroup$ – Harch Oct 13 '13 at 18:28
  • $\begingroup$ @Harch: That's worth posting as a new question. It's not trivial. $\endgroup$ – John Bentin Oct 14 '13 at 9:23
  • $\begingroup$ Ok I have posted the follow up question. Just out of interest what is your background? (The only reason I am asking is because this is a rare speciality on RMT and I am finding it difficult to find like minded people) $\endgroup$ – Harch Oct 16 '13 at 9:36
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    $\begingroup$ @Harch: I'm no expert in number theory or complex analysis, just an amateur, but like many others I keep in touch with developments in research on the non-trivial zeros. Mostly the results are empirical, observing statistical patterns in the huge number of zeros that have been plotted up till now. As far as I know, this result is empirical too. You will need to find someone else to get more information. $\endgroup$ – John Bentin Oct 16 '13 at 12:22
  • $\begingroup$ can be another function with a mean density of zeros equal to $\frac{t}{2\pi}log \frac{t}{2\pi} $ which is completely different from the riemann zeta function ·$ \zeta (s) $ $\endgroup$ – Jose Garcia Nov 28 '16 at 17:55

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