We study moments of the logarithmic derivative of characteristic polynomials of orthogonal and symplectic random matrices. In particular, we compute the asymptotics for large matrix size, N, of these moments evaluated at points that are approaching 1. This follows the work of Bailey et al. [J. Math. Phys. 60(8), 083509 (2019)] where they computed these asymptotics in the case of unitary random matrices.
REFERENCES
1.
E. C.
Bailey
, S.
Bettin
, G.
Blower
, J. B.
Conrey
, A.
Prokhorov
, M. O.
Rubinstein
, and N. C.
Snaith
, “Mixed moments of characteristic polynomials of random unitary matrices
,” J. Math. Phys.
60
(8
), 083509
(2019
); arXiv:math.nt/1901.07479.2.
J. B.
Conrey
, D. W.
Farmer
, J. P.
Keating
, M. O.
Rubinstein
, and N. C.
Snaith
, “Integral moments of L-functions
,” Proc. London Math. Soc.
91
(1
), 33
–104
(2005
).3.
C. P.
Hughes
, “Random matrix theory and discrete moments of the Riemann zeta function
,” J. Phys. A: Math. Gen.
36
(12
), 2907
–2917
(2003
).4.
J. P.
Keating
and N. C.
Snaith
, “Random matrix theory and ζ(1/2 + it)
,” Commun. Math. Phys.
214
, 57
–89
(2000
).5.
J. B.
Conrey
, “L-functions and random matrices
,” in Mathematics Unlimited 2001 and Beyond
, edited by B.
Enquist
and W.
Schmid
(Springer-Verlag
, Berlin
, 2001
, pp. 331
–352
; arXiv:math.nt/0005300.6.
J. P.
Keating
and N. C.
Snaith
, “Random matrices and L-functions
,” J. Phys. A: Math. Gen.
36
(12
), 2859
–2881
(2003
).7.
N. C.
Snaith
, “Riemann zeros and random matrix theory
,” Milan J. Math.
78
(1
), 135
–152
(2010
) [Conference: 1st School of the Riemann-International-School-of-Mathematics, Verbania, Italy
].8.
J. B.
Conrey
and D. W.
Farmer
, “Mean values of L-functions and symmetry
,” Int. Math. Res. Not.
2000
(17
), 883
–908
; arXiv:math.nt/9912107.9.
B.
Conrey
, D. W.
Farmer
, and M. R.
Zirnbauer
, “Autocorrelation of ratios of L-functions
,” Commun. Number Theory Phys.
2
(3
), 593
–636
(2008
); arXiv:0711.0718.10.
N. M.
Katz
and P.
Sarnak
, Random Matrices, Frobenius Eigenvalues and Monodromy
(AMS Colloquium Publications
, Providence
, 1998
).11.
N. M.
Katz
and P.
Sarnak
, “Zeros of zeta functions and symmetry
,” Bull. Am. Math. Soc.
36
, 1
–26
(1999
).12.
J. P.
Keating
and N. C.
Snaith
, “Random matrix theory and L-functions at s = 1/2
,” Commun. Math. Phys.
214
, 91
–110
(2000
).13.
A.
Selberg
, “On the normal density of primes in small intervals, and the difference between consecutive primes
,” Arch. Math. Nat.
47
(6
), 87
–105
(1943
).14.
D. A.
Goldston
, S. M.
Gonek
, and H. L.
Montgomery
, “Mean values of the logarithmic derivative of the Riemann zeta-function with applications to primes in short intervals
,” J. Reine Angew Math.
2001
(537
), 105
–126
.15.
D. W.
Farmer
, S. M.
Gonek
, Y.
Lee
, and S. J.
Lester
, “Mean values of ζ′/ζ(s), correlations of zeros and the distribution of almost primes
,” Q. J. Math.
64
(4
), 1057
–1089
(2013
).16.
J.
Conrey
and N.
Snaith
, “Correlations of eigenvalues and Riemann zeros
,” Commun. Number Theory Phys.
2
(3
), 477
–536
(2008
).17.
D. W.
Farmer
, “Mean values of ζ′/ζ and the Gaussian unitary ensemble hypothesis
,” Int. Math. Res. Not.
1995
(2
), 71
–82
.18.
C. R.
Guo
, “The distribution of the logarithmic derivative of the Riemann zeta function
,” Proc. London Math Soc.
72
(3
), 1
–27
(1996
).19.
S. J.
Lester
, “The distribution of the logarithmic derivative of the Riemann zeta-function
,” Q. J. Math.
65
, 1319
–1344
(2014
).20.
C. P.
Hughes
, J. P.
Keating
, and N.
O’Connell
, “Random matrix theory and the derivative of the Riemann zeta function
,” Proc. R. Soc. London, Ser. A
456
, 2611
–2627
(2000
).21.
F.
Mezzadri
, “Random matrix theory and the zeroes of ζ′(s)
,” J. Phys. A: Math. Gen.
36
(12
), 2945
–2962
(2003
).22.
J. B.
Conrey
, M. O.
Rubinstein
, and N. C.
Snaith
, “Moments of the derivative of characteristic polynomials with an application to the Riemann zeta function
,” Commun. Math. Phys.
267
(3
), 611
–629
(2006
).23.
C. P.
Hughes
, “On the characteristic polynomial of a random unitary matrix and the Riemann zeta function
,” Ph.D. thesis, University of Bristol
, 2001
.24.
P.-O.
Dehaye
, “Joint moments of derivatives of characteristic polynomials
,” Alg. Number Theory
2
(1
), 31
–68
(2008
).25.
H.
Riedtmann
, “A combinatorial approach to mixed ratios of characteristic polynomials
,” arXiv:math.nt/1805.07261 (2018
).26.
B.
Winn
, “Derivative moments for characteristic polynomials from the CUE
,” Commun. Math. Phys.
315
, 531
–562
(2012
).27.
J.
Conrey
, P.
Forrester
, and N.
Snaith
, “Averages of ratios of characteristic polynomials for the compact classical groups
,” Int. Math. Res. Not.
2005
(7
), 397
–431
.28.
A. M.
Mason
and N. C.
Snaith
, , Memoirs of the American Mathematical Society (AMS
, 2018
), Vol. 251, No. 1194.© 2020 Author(s).
2020
Author(s)
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