# Mean density of the nontrivial zeros of the Riemann zeta function

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 $t$.
• 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 ·$\zeta (s)$ – Jose Garcia Nov 28 '16 at 17:55