Theorem - If the Riemann hypothesis would be true, and the Montgomery pair correlation conjecture (see linked article page 183-184) true too; let $p \in \Bbb P$ prime, $n \in \Bbb N$ and

$$\mathcal V_p(n)=1-(n^{p-1}\mod p)$$

then$$\lim_{p\rightarrow \infty}\mathcal V_p(n) = \operatorname{sinc}(2\pi \,n)$$

and for the pair correlation of the non-trivial zeroes of the Riemann $\zeta$-function, holds

$$r_2(u)=1-\operatorname{sinc^2}(\pi \,u) +\delta(u)=1-(\lim_{p\rightarrow \infty}\mathcal V_p)^2 +\delta(u)$$

with $u=2n$ (even).

$\delta(u)$ stands for the Dirac $\delta$-function, $r_2(u)$ denotes the pair correlation function, describing the two point correlation of the non-trivial zeros of the Riemann $\zeta$-function.

Question: Having this theorem settled, in which different ways may we interprete it intuitively?

I really beg you for interprations from probably different viewpoints but helping for a better comprehension of $r_2(u)$ versus $\mathcal V_p$ (and vice versa).

Marginal note: The theorem applies generally (in regard to random matrix theory) to the sine-kernel of local level correlation statistics for Gaussian Unitary Ensembles (Gaudin, Dyson, Mehta) and corresponding level repulsion.

  • $\begingroup$ Your $V_p$ is actually a function of $n$, $V_p(n)$, and it's 1 if $p$ divides $n$, zero otherwise. So for any fixed $n$, $\lim_{p\to\infty}V_p(n)=0$. Also, you use $r_2$ and $\delta$ without defining either of them. $\endgroup$ – Gerry Myerson Jul 7 '13 at 10:27

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