# Triple correlation of the Riemann zeros

• [1] American Institute of Mathematics 360 Portage Ave Palo Alto, CA 94306 School of Mathematics University of Bristol Bristol, BS8 1TW, United Kingdom
• [2] School of Mathematics University of Bristol Bristol, BS8 1TW, United Kingdom
• Volume: 20, Issue: 1, page 61-106
• ISSN: 1246-7405

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## Abstract

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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].

## How to cite

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Conrey, J. Brian, and Snaith, Nina C.. "Triple correlation of the Riemann zeros." Journal de Théorie des Nombres de Bordeaux 20.1 (2008): 61-106. <http://eudml.org/doc/10833>.

@article{Conrey2008,
abstract = {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 $U(N)$ 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].},
affiliation = {American Institute of Mathematics 360 Portage Ave Palo Alto, CA 94306 School of Mathematics University of Bristol Bristol, BS8 1TW, United Kingdom; School of Mathematics University of Bristol Bristol, BS8 1TW, United Kingdom},
author = {Conrey, J. Brian, Snaith, Nina C.},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {zeros of the Riemann zeta function; random matrices},
language = {eng},
number = {1},
pages = {61-106},
publisher = {Université Bordeaux 1},
title = {Triple correlation of the Riemann zeros},
url = {http://eudml.org/doc/10833},
volume = {20},
year = {2008},
}

TY - JOUR
AU - Conrey, J. Brian
AU - Snaith, Nina C.
TI - Triple correlation of the Riemann zeros
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2008
PB - Université Bordeaux 1
VL - 20
IS - 1
SP - 61
EP - 106
AB - 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 $U(N)$ 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 - eng
KW - zeros of the Riemann zeta function; random matrices
UR - http://eudml.org/doc/10833
ER -

## References

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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. Zbl1075.11058MR2149530
11. J.B. Conrey, D.W. Farmer, M.R. Zirnbauer, Autocorrelation of ratios of characteristic polynomials and of $L$-functions. arXiv:0711.0718. Zbl1178.11056
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