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Montgomery's pair correlation conjecture

Montgomery's pair correlation conjecture is a conjecture made by (Montgomery 1973).

Conjecture (Montgomery's pair correlation conjecture, 1973). (Montgomery)

The pair correlation function ${\displaystyle \scriptstyle R_{2}(u)\,}$ between pairs of adjacent nontrivial zeros (assumed to be on the critical line) of the Riemann zeta function (normalized to have unit average spacing) is
${\displaystyle R_{2}(u)=1-\mathrm {sinc} ^{2}\,u+\delta (u)=1-\left({\frac {\sin \pi u}{\pi u}}\right)^{2}+\delta (u).\,}$

As Freeman Dyson pointed out to him, it is the same as the pair correlation function of random Hermitian matrices.

Informally, this means that the chance of finding a zero in a very short interval of length ${\displaystyle \scriptstyle {\frac {2\pi L}{\log T}}\,}$ at a distance ${\displaystyle \scriptstyle {\frac {2\pi u}{\log T}}\,}$ from a zero ${\displaystyle \scriptstyle {\frac {1}{2}}+iT\,}$ is about ${\displaystyle \scriptstyle L\,}$ times the expression above. (The factor ${\displaystyle \scriptstyle {\frac {2\pi }{\log T}}\,}$ is a normalization factor that can be thought of informally as the average spacing between zeros with imaginary part about ${\displaystyle \scriptstyle T\,}$.) (Odlyzko 1987) showed that the conjecture was supported by large-scale computer calculations of the zeros. The conjecture has been extended to correlations of more than 2 zeros, and also to zeta functions of automorphic representations (Rudnick & Sarnak 1996).

Montgomery was studying the Fourier transform ${\displaystyle \scriptstyle F(x)\,}$ of the pair correlation function, and showed (assuming the Riemann hypothesis) that it was equal to ${\displaystyle \scriptstyle |x|\,}$ for ${\displaystyle \scriptstyle |x|\,<\,1\,}$. His methods were unable to determine it for ${\displaystyle \scriptstyle |x|\,\geq \,1\,}$, but he conjectured that it was equal to 1 for these ${\displaystyle \scriptstyle x\,}$, which implies that the pair correlation function is as above.

1 - ((sin pi x)/(pi x))^2

Maclaurin series expansion of 1 - ((sin pi x)/(pi x))^2

The Maclaurin series expansion of 1 - ((sin pi x)/(pi x))^2 is

${\displaystyle 1-\mathrm {sinc} _{\pi }^{2}\,x=1-\left({\frac {\sin \pi x}{\pi x}}\right)^{2}={\frac {\pi ^{2}x^{2}}{3}}-{\frac {2\pi ^{4}x^{4}}{45}}+{\frac {\pi ^{6}x^{6}}{315}}-{\frac {2\pi ^{8}x^{8}}{14175}}+{\frac {2\pi ^{10}x^{10}}{467775}}-{\frac {4\pi ^{12}x^{12}}{42567525}}+{\frac {\pi ^{14}x^{14}}{638512875}}-{\frac {2\pi ^{16}x^{16}}{97692469875}}+{\frac {2\pi ^{18}x^{18}}{9280784638125}}-\cdots ,\,}$

where ${\displaystyle \scriptstyle \mathrm {sinc} _{\pi }\,x\,}$ is the normalized sinc function.

1 - ((sin x)/x)^2

The Maclaurin series expansion of 1 - sinc^2 x = 1 - ((sin x)/x)^2 is

${\displaystyle 1-\mathrm {sinc} ^{2}\,x=1-\left({\frac {\sin x}{x}}\right)^{2}={\frac {x^{2}}{3}}-{\frac {2x^{4}}{45}}+{\frac {x^{6}}{315}}-{\frac {2x^{8}}{14175}}+{\frac {2x^{10}}{467775}}-{\frac {4x^{12}}{42567525}}+{\frac {x^{14}}{638512875}}-{\frac {2x^{16}}{97692469875}}+{\frac {2x^{18}}{9280784638125}}-{\frac {4x^{20}}{2143861251406875}}+\cdots ,\,}$

where ${\displaystyle \scriptstyle \mathrm {sinc} \,x\,}$ is the unnormalized sinc function.

Numerators of Maclaurin series for 1 - ((sin x)/x)^2

A048896 2A000120(n+1) - 1, ${\displaystyle \scriptstyle n\,\geq \,1\,}$. Maximal power of 2 dividing ${\displaystyle \scriptstyle n\,}$ th Catalan number (A000108).

{1, 2, 1, 2, 2, 4, 1, 2, 2, 4, 2, 4, 4, 8, 1, 2, 2, 4, 2, 4, 4, 8, 2, 4, 4, 8, 4, 8, 8, 16, 1, 2, 2, 4, 2, 4, 4, 8, 2, 4, 4, 8, 4, 8, 8, 16, 2, 4, 4, 8, 4, 8, 8, 16, 4, 8, 8, 16, 8, 16, 16, 32, 1, 2, 2, 4, 2, 4, 4, 8, 2, 4, 4, 8, 4, 8, 8, 16, 2, 4, 4, 8, 4, 8, 8, 16, 4, 8, 8, 16, 8, 16, 16, 32, 2, 4, 4, ...}

A000120 1's-counting sequence: number of 1's in binary expansion of ${\displaystyle \scriptstyle n\,}$ (or the binary weight of ${\displaystyle \scriptstyle n,\,n\,\geq \,2\,}$.)

{1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 3, 4, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4, 5, 5, 6, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4, ...}

Denominators of Maclaurin series for 1 - ((sin x)/x)^2

A117972 Numerator of ${\displaystyle \scriptstyle \zeta '(-2n),\,n\,\geq \,2\,}$.

{3, -45, 315, -14175, 467775, -42567525, 638512875, -97692469875, 9280784638125, -2143861251406875, 147926426347074375, -48076088562799171875, 9086380738369043484375, -3952575621190533915703125, ...}