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This question will be based roughly on the Bourgade Keating review on Zeta function and eigenvalue asymptotics (BK): https://link.springer.com/chapter/10.1007/978-3-0348-0697-8_4

To set up the question, we consider the riemann zeta function $\zeta(s)$ with zeros on the critical line $\frac{1}{2} + i t_n$. The unfolded zeros are defined as: $w_n = \frac{t_n}{2\pi} \log \frac{t_n}{2\pi}$. The name "unfolded" is justified by the fact that distribution of $w_n$ is asymptotically uniform along the critical axis. In terms of these unfolded zeros, we can define the following integral: \begin{equation} R_{2,\zeta}(f,W) = \int_{-\infty}^{\infty} f(x) \frac{1}{W} \sum_{\substack {j\neq k \\ w_j,w_k \leq W}} \delta(x-w_j+w_k) dx \end{equation}

Suppose the $w_n$'s had no pair correlation, the integral above would simply equal to $\int f(x) dx$. However, $w_n$'s clearly have pair correlations! So deviation of the integral from $\int f(x) dx$ roughly measures the pair correlations between the unfolded zeros $w_n$, up to some cutoff $W$. For $\zeta$ functions Montgomery proved an important theorem about this integral stated in KS as follows:

Theorem 1 (Montgomery): Assume the Riemann Hypothesis. Then for test functions f(x) such that: $$ \hat f(\tau) = \int_{-\infty}^{\infty} e^{2 \pi i x \tau} f(x) dx$$ has support in $(-1,1)$, the following limit exists: $$ \lim_{W \rightarrow \infty} R_{2,\zeta}(f,W) = \int_{-\infty}^{\infty} f(x) R_2(x) dx $$ with $$R_2(x) = 1 - (\frac{\sin(\pi x)}{\pi x})^2 $$

The confusion I have is: this $R_2(x)$ seems to carry unnecessary information. To show that, suppose I Fourier transform the integral: \begin{equation} \int_{-\infty}^{\infty} f(x) R_2(x) dx = \int_{-\infty}^{\infty} \hat f(\tau) K(-\tau) d \tau \end{equation} Where $K(\tau)$ is the Fourier transform of the pair correlation function, sometimes referred to as the spectral form factor. Then since $\hat f(\tau)$ only has support on $(-1,1)$, I can restrict the $\tau$ integral to $(-1,1)$. But that would mean the integral is only sensitive to $K(\tau)$ for $\tau \in (-1,1)$. Within that domain, there are other functions that produce the same $K(\tau)$. For example, following equation (26) in BK, we can define: $$ \tilde R_2(x) = 1 - \frac{1}{2 (\pi x)^2} \quad R_2(x) = 1 - (\frac{\sin(\pi x)}{\pi x})^2 $$ These functions have the same Fourier transform within $(-1,1)$.

Montgomery's Theorem is usually stated as a connection between random matrix theory and zeta function because $R_2(x)$ is precisely the pair correlation of Wigner random matrix ensembles. However, the calculation above would suggest that $\tilde R_2(x)$ would do just as well within the domain of interest. Hence, I feel that $R_2(x)$ carries unnecessary information about the pair correlations, and it now seems rather artificial that it matches the random matrix results.

This seems like a simple enough question. But I haven't found any explanation of it in various review articles on RMT-zeta function connections. So I would like some help from the experts: Why did Montgomery put the RMT correlation function in his theorem if it carries unnecessary information? Was it just an inspired guess or is there something deep I am missing?

Note: The referenced review article seem to be on the boundary between math and physics. But since my question is more about the mathematical side, I thought it would be best to pose it here. If the moderators feel this is more fitting for other Physics SE or Math SE, please help me move the question to the right place. Thanks

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Assuming the Riemann Hypothesis Montgomery consider the function $$F(\alpha)=F(\alpha,T)=\Bigl(\frac{T}{2\pi}\log T\Bigr)^{-1} \sum_{0<\gamma, \gamma'\le T} T^{i\alpha(\gamma-\gamma')}w(\gamma-\gamma')$$ where $w(x)=4/(4+x^2)$.

He proved results only in the range $-1+\varepsilon\le\alpha\le 1-\varepsilon$.
Therefore he restricted $\widehat r(\alpha)$ to this range and asserts $$\sum_{0\le \gamma,\gamma'\le T} r\Bigl((\gamma-\gamma')\frac{\log T}{2\pi}\Bigr) w(\gamma-\gamma')=\Bigl(\frac{T}{2\pi}\log T\Bigr)\int_{-\infty}^{+\infty}F(\alpha)\widehat{r}(\alpha).$$ From this result he obtains some consequences. For example, at least $2/3$ of the zeros are simple, that are proved conditionally on the Riemann hypothesis.

Then he give reasons to assume that $F(\alpha)\sim1$ for $|\alpha|\ge1$ and make his conjecture that $1-((\sin\pi u)/\pi u)^2$ is the pair correlation function for the zeros of the zeta function. This leads then to F. J. Dyson famous remarks.

So, you have to distinguish between what Montgomery proved under Riemann hypothesis and the conjecture of the correlation that goes beyond the Riemann hypothesis.

Montgomery's conjecture was reinforced by the statistics of Odlyzko. There is also another alternative hypothesis which postulates the normalized zeros tends to be separated by integers or half integers.

Odlyzko's figure was restricted to the range $0\le \gamma-\gamma'\le 6$. In 2013 I duplicated this statistics but using the range $0\le \gamma-\gamma'\le 40$. My figures revealed a surprising structure for this range, more in the mood of the alternative hypothesis.

Extended Odlyzko statistics

I used Pratt zeros for this statistics, nevertheless it will be useful that somebody with more computer expertise than I cofirm my results. The figure is obtained from the zeros in the range 20.000.000 to 30.000.000 more or less.

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  • $\begingroup$ It is true that when I used zeros on the range $n=103 752 000 248 the figure was near the one predicted by Montgomery conjecture. $\endgroup$ – juan Jul 1 '18 at 9:45
  • $\begingroup$ More specifically, Montgomery conjectured that $F(\alpha)$ should be $1$ for $|\alpha|$ between $1$ and $2$ based on the twin primes conjecture, and that it should be close to $1$ on average based on the overall count of zeroes. This led to the hypothesis that it is $1$ for $|\alpha|=1$. This is explained, briefly but straightforwardly, in Montgomery's original paper. $\endgroup$ – Will Sawin Jul 2 '18 at 7:15

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