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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 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

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 at a distance from a zero is about times the expression above. (The factor is a normalization factor that can be thought of informally as the average spacing between zeros with imaginary part about .) (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 of the pair correlation function, and showed (assuming the Riemann hypothesis) that it was equal to for . His methods were unable to determine it for , but he conjectured that it was equal to 1 for these , 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

where 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

where is the unnormalized sinc function.

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

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

{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 .

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

See also



References

  • 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 ..

External links