Triple correlation of the Riemann zeros
Journal de théorie des nombres de Bordeaux, Volume 20 (2008) no. 1, p. 61-106

We use the conjecture of Conrey, Farmer and Zirnbauer for averages of ratios of the Riemann zeta function [11] to calculate all the lower order terms of the triple correlation function of the Riemann zeros. A previous approach was suggested by Bogomolny and Keating [6] taking inspiration from semi-classical methods. At that point they did not write out the answer explicitly, so we do that here, illustrating that by our method all the lower order terms down to the constant can be calculated rigourously if one assumes the ratios conjecture of Conrey, Farmer and Zirnbauer. Bogomolny and Keating [4] returned to their previous results simultaneously with this current work, and have written out the full expression. The result presented in this paper agrees precisely with their formula, as well as with our numerical computations, which we include here.

We also include an alternate proof of the triple correlation of eigenvalues from random $U\left(N\right)$ matrices which follows a nearly identical method to that for the Riemann zeros, but is based on the theorem for averages of ratios of characteristic polynomials [12, 13].

La conjecture de Conrey, Farmer et Zirnbauer [11] concernant les moyennes de quotients de la fonction $\zeta$ de Riemann nous permet de déterminer tous les termes d’ordre inférieur de la corrélation triple des zéros de la fonction $\zeta$ de Riemann. Bogomolny et Keating [4], en s’inspirant de méthodes semi-classiques, avaient suggéré en 1996 une approche permettant de s’attaquer à ce problème mais n’avaient pas donné explicitement le résultat qui est l’objet de cet article. Notre méthode permet de déterminer rigoureusement tous les termes d’ordre inférieur jusqu’au terme constant en admettant les conjectures de quotients de Conrey, Farmer et Zirnbauer. Bogomolny et Keating [4] se sont de nouveau penchés sur leurs résultas en même temps que ce travail et ont donné l’expression entière explicite. Le résultat décrit dans cet article coincide exactement avec leur formule et est en accord avec nos calculs numériques qui sont aussi présentés ici.

Nous donnons également une autre preuve de la corrélation triple des valeurs propres des matrices aléatoires $U\left(N\right)$ qui suit une méthode quasiment identique à celle employée pour les zéros de la fonction $\zeta$ de Riemann mais qui repose sur le théorème concernant les moyennes des quotients de polynômes caractéristiques [12, 13].

@article{JTNB_2008__20_1_61_0,
author = {Conrey, J. Brian and Snaith, Nina C.},
title = {Triple correlation of the Riemann zeros},
journal = {Journal de th\'eorie des nombres de Bordeaux},
publisher = {Universit\'e Bordeaux 1},
volume = {20},
number = {1},
year = {2008},
pages = {61-106},
doi = {10.5802/jtnb.616},
mrnumber = {2434158},
zbl = {pre05543191},
language = {en},
url = {http://www.numdam.org/item/JTNB_2008__20_1_61_0}
}
Conrey, J. Brian; Snaith, Nina C. Triple correlation of the Riemann zeros. Journal de théorie des nombres de Bordeaux, Volume 20 (2008) no. 1, pp. 61-106. doi : 10.5802/jtnb.616. http://www.numdam.org/item/JTNB_2008__20_1_61_0/

[1] M.V. Berry, Semiclassical formula for the number variance of the Riemann zeros. Nonlinearity 1 (1988), 399–407. | MR 955621 | Zbl 0664.10022

[2] M.V. Berry, J.P. Keating, The Riemann zeros and eigenvalue asymptotics. SIAM Rev. 41(2) (1999), 236–266. | MR 1684543 | Zbl 0928.11036

[3] E.B. Bogomolny, Spectral statistics and periodic orbits. In New Directions in Quantum Chaos; editors, G. Casati, I. Guarneri, and U. Smilansky, pages 333–369. IOS Press, Amsterdam, 2000. | MR 1843514 | Zbl 1001.81027

[4] E.B. Bogomolny, J.P. Keating, private communication.

[5] E.B. Bogomolny, J.P. Keating, Random matrix theory and the Riemann zeros I: three- and four-point correlations. Nonlinearity 8 (1995), 1115–1131. | MR 1363402 | Zbl 0834.60105

[6] E.B. Bogomolny, J.P. Keating, Gutzwiller’s trace formula and spectral statistics: beyond the diagonal approximation. Phys. Rev. Lett. 77(8) (1996), 1472–1475.

[7] E.B. Bogomolny, J.P. Keating, Random matrix theory and the Riemann zeros II:$n$-point correlations. Nonlinearity 9 (1996), 911–935. | MR 1399479 | Zbl 0898.60107

[8] J.B. Conrey, $L$-functions and random matrices. In Mathematics Unlimited 2001 and Beyond; editors, B. Enquist and W. Schmid, pages 331–352. Springer-Verlag, Berlin, 2001. arXiv:math.nt/0005300. | MR 1852163 | Zbl 1048.11071

[9] J.B. Conrey, Notes on eigenvalue distributions for the classical compact groups. In Recent perspectives on random matrix theory and number theory, LMS Lecture Note Series 322, pages 111–45. Cambridge University Press, Cambridge, 2005. | MR 2166460 | Zbl pre05046196

[10] J.B. Conrey, D.W. Farmer, J.P. Keating, M.O. Rubinstein, N.C. Snaith, Integral moments of $L$-functions. Proc. Lond. Math. Soc. 91 No. 1 (2005), pages 33–104. arXiv:math.nt/0206018. | MR 2149530 | Zbl 1075.11058

[11] J.B. Conrey, D.W. Farmer, M.R. Zirnbauer, Autocorrelation of ratios of characteristic polynomials and of $L$-functions. arXiv:0711.0718.

[12] J.B. Conrey, D.W. Farmer, M.R. Zirnbauer, Howe pairs, supersymmetry, and ratios of random characteristic polynomials for the unitary groups $U\left(N\right)$. Preprint. arXiv:math-ph/0511024.

[13] J.B. Conrey, P.J. Forrester, N.C. Snaith, Averages of ratios of characteristic polynomials for the compact classical groups. Int. Math. Res. Notices 7 (2005), 397–431. | MR 2130839 | Zbl pre02162778

[14] J.B. Conrey, N.C. Snaith, Applications of the $L$-functions ratios conjectures. Proc. Lon. Math. Soc. 94(3) (2007), 594–646. arXiv:math.NT/0509480. | MR 2325314 | Zbl pre05166583

[15] J.B. Conrey, N.C.Snaith, Correlations of eigenvalues and Riemann zeros. Preprint. arXiv:0803.2795.

[16] G.H. Hardy, J.E. Littlewood, Contributions to the theory of the Riemann zeta-function and the theory of the distribution of primes. Acta Mathematica 41 (1918), 119–196. | MR 1555148

[17] D.A. Hejhal, On the triple correlation of zeros of the zeta function. Inter. Math. Res. Notices 7 (1994), 293–302. | MR 1283025 | Zbl 0813.11048

[18] J.P. Keating, The Riemann zeta function and quantum chaology. In Quantum Chaos; editors, G. Casati, I Guarneri, and U. Smilansky, pages 145–85. North-Holland, Amsterdam, 1993. | MR 1246830

[19] J.P. Keating, Periodic orbits, spectral statistics, and the Riemann zeros. In Supersymmetry and trace formulae: chaos and disorder; editors, I.V. Lerner, J.P. Keating, and D.E. Khmelnitskii, pages 1–15. Plenum, New York, 1999.

[20] J.P. Keating, N.C. Snaith, Random matrix theory and $\zeta \left(1/2+it\right)$. Commun. Math. Phys. 214 (200), 57–89. | MR 1794265 | Zbl 1051.11048

[21] J.P. Keating, N.C. Snaith, Random matrices and $L$-functions. J. Phys. A 36(12) (2003), 2859–81. | MR 1986396 | Zbl 1074.11053

[22] M.L. Mehta, Random Matrices. Academic Press, London, second edition, 1991. | MR 1083764 | Zbl 0780.60014

[23] H.L. Montgomery, The pair correlation of the zeta function. Proc. Symp. Pure Math 24 (1973), 181–93. | MR 337821 | Zbl 0268.10023

[24] Z. Rudnick, P. Sarnak, Zeros of principal $L$-functions and random matrix theory. Duke Mathematical Journal 81(2) (1996), 269–322. | MR 1395406 | Zbl 0866.11050