Triple correlation of the Riemann zeros
Journal de Théorie des Nombres de Bordeaux, Volume 20 (2008) no. 1, pp. 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].

Published online:
DOI: 10.5802/jtnb.616
J. Brian Conrey 1, 2; Nina C. Snaith 2

1 American Institute of Mathematics 360 Portage Ave Palo Alto, CA 94306
2 School of Mathematics University of Bristol Bristol, BS8 1TW, United Kingdom
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J. Brian Conrey; Nina C. Snaith. 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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.616/

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