Montgomery's pair correlation conjecture

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

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

The pair correlation function $\scriptstyle R_{2}(u)\,$ 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

$R_{2}(u)=1-\mathrm {sinc} ^{2}\,u+\delta (u)=1-\left({\frac {\sin \pi u}{\pi u}}\right)^{2}+\delta (u).\,$

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 $\scriptstyle {\frac {2\pi L}{\log T}}\,$ at a distance $\scriptstyle {\frac {2\pi u}{\log T}}\,$ from a zero $\scriptstyle {\frac {1}{2}}+iT\,$ is about $\scriptstyle L\,$ times the expression above. (The factor $\scriptstyle {\frac {2\pi }{\log T}}\,$ is a normalization factor that can be thought of informally as the average spacing between zeros with imaginary part about $\scriptstyle T\,$.) (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 $\scriptstyle F(x)\,$ of the pair correlation function, and showed (assuming the Riemann hypothesis) that it was equal to $\scriptstyle |x|\,$ for $\scriptstyle |x|\,<\,1\,$. His methods were unable to determine it for $\scriptstyle |x|\,\geq \,1\,$, but he conjectured that it was equal to 1 for these $\scriptstyle x\,$, 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

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

where $\scriptstyle \mathrm {sinc} _{\pi }\,x\,$ 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

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

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

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

A048896 2 A000120(n+1) - 1, $\scriptstyle n\,\geq \,1\,$. Maximal power of 2 dividing $\scriptstyle n\,$ 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 $\scriptstyle n\,$ (or the binary weight of $\scriptstyle n,\,n\,\geq \,2\,$.)

{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 $\scriptstyle \zeta '(-2n),\,n\,\geq \,2\,$.

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

See also

• Riemann zeta function
• Riemann zeta function nontrivial zeros

• Sinc function

• Montgomery-Odlyzko law
• Hilbert-Pólya conjecture

References

• Katz, Nicholas M.; Sarnak, Peter (1999), “Zeroes of zeta functions and symmetry”, American Mathematical Society. Bulletin. New Series 36 (1): 1–26, doi:10.1090/S0273-0979-99-00766-1, ISSN 0002-9904, http://www.ams.org/bull/1999-36-01/S0273-0979-99-00766-1/home.html ..

• Montgomery, Hugh L. (1973), “The pair correlation of zeros of the zeta function”, Analytic number theory, Proc. Sympos. Pure Math., XXIV, Providence, R.I.: American Mathematical Society, pp. 181–193 ..

• Odlyzko, Andrew M. (1987), “On the distribution of spacings between zeros of the zeta function”, Mathematics of Computation (American Mathematical Society) 48 (177): 273–308, doi:10.2307/2007890, ISSN 0025-5718 ..

• Rudnick, Zeév; Sarnak, Peter (1996), “Zeros of principal L-functions and random matrix theory”, Duke Mathematical Journal 81 (2): 269–322, doi:10.1215/S0012-7094-96-08115-6, ISSN 0012-7094, http://projecteuclid.org/euclid.dmj/1077245671 ..

External links

• Pei Li, On Montgomery's pair correlation conjecture to the zeros of Riemann zeta function, Thesis (M.Sc.), Department of Mathematics - Simon Fraser University, 2005.
• G. D. Mostow, On a Conjecture of Montgomery, The Annals of Mathematics, Second Series, Vol. 65, No. 3, May, 1957.