×

Moments of the logarithmic derivative of characteristic polynomials from \(SO (N)\) and \(USp (2N)\). (English) Zbl 1454.15022

Summary: 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 E. C. Bailey et al. [J. Math. Phys. 60, No. 8, 083509, 26 p. (2019; Zbl 1476.15054)] where they computed these asymptotics in the case of unitary random matrices.
©2020 American Institute of Physics

MSC:

15B52 Random matrices (algebraic aspects)
15A18 Eigenvalues, singular values, and eigenvectors
60B20 Random matrices (probabilistic aspects)

Citations:

Zbl 1476.15054
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Bailey, E. C.; Bettin, S.; Blower, G.; Conrey, J. B.; Prokhorov, A.; Rubinstein, M. O.; Snaith, N. C., Mixed moments of characteristic polynomials of random unitary matrices, J. Math. Phys., 60, 8, 083509 (2019) · Zbl 1476.15054
[2] Conrey, J. B.; Farmer, D. W.; Keating, J. P.; Rubinstein, M. O.; Snaith, N. C., Integral moments of L-functions, Proc. London Math. Soc., 91, 1, 33-104 (2005) · Zbl 1075.11058
[3] Hughes, C. P., Random matrix theory and discrete moments of the Riemann zeta function, J. Phys. A: Math. Gen., 36, 12, 2907-2917 (2003) · Zbl 1074.11047
[4] Keating, J. P.; Snaith, N. C., Random matrix theory and ζ(1/2 + it), Commun. Math. Phys., 214, 57-89 (2000) · Zbl 1051.11048
[5] Conrey, J. B.; Enquist, B.; Schmid, W., L-functions and random matrices, Mathematics Unlimited 2001 and Beyond, 331-352 (2001), Springer-Verlag: Springer-Verlag, Berlin · Zbl 1048.11071
[6] Keating, J. P.; Snaith, N. C., Random matrices and L-functions, J. Phys. A: Math. Gen., 36, 12, 2859-2881 (2003) · Zbl 1074.11053
[7] Snaith, N. C., Riemann zeros and random matrix theory, Milan J. Math., 78, 1, 135-152 (2010) · Zbl 1310.11089
[8] Conrey, J. B.; Farmer, D. W., Mean values of L-functions and symmetry, Int. Math. Res. Not., 2000, 17, 883-908 · Zbl 1035.11038
[9] Conrey, B.; Farmer, D. W.; Zirnbauer, M. R., Autocorrelation of ratios of L-functions, Commun. Number Theory Phys., 2, 3, 593-636 (2008) · Zbl 1178.11056
[10] Katz, N. M.; Sarnak, P., Random Matrices, Frobenius Eigenvalues and Monodromy (1998), AMS Colloquium Publications: AMS Colloquium Publications, Providence
[11] Katz, N. M.; Sarnak, P., Zeros of zeta functions and symmetry, Bull. Am. Math. Soc., 36, 1-26 (1999) · Zbl 0921.11047
[12] Keating, J. P.; Snaith, N. C., Random matrix theory and L-functions at s = 1/2, Commun. Math. Phys., 214, 91-110 (2000) · Zbl 1051.11047
[13] Selberg, A., On the normal density of primes in small intervals, and the difference between consecutive primes, Arch. Math. Nat., 47, 6, 87-105 (1943) · Zbl 0063.06869
[14] Goldston, D. A.; Gonek, S. M.; Montgomery, H. L., 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 · Zbl 0984.11044
[15] Farmer, D. W.; Gonek, S. M.; Lee, Y.; Lester, S. J., Mean values of ζ^′/ζ(s), correlations of zeros and the distribution of almost primes, Q. J. Math., 64, 4, 1057-1089 (2013) · Zbl 1297.11099
[16] Conrey, J.; Snaith, N., Correlations of eigenvalues and Riemann zeros, Commun. Number Theory Phys., 2, 3, 477-536 (2008) · Zbl 1169.11035
[17] Farmer, D. W., Mean values of ζ^′/ζ and the Gaussian unitary ensemble hypothesis, Int. Math. Res. Not., 1995, 2, 71-82 · Zbl 0829.11043
[18] Guo, C. R., The distribution of the logarithmic derivative of the Riemann zeta function, Proc. London Math Soc., 72, 3, 1-27 (1996) · Zbl 0833.11042
[19] Lester, S. J., The distribution of the logarithmic derivative of the Riemann zeta-function, Q. J. Math., 65, 1319-1344 (2014) · Zbl 1317.11086
[20] Hughes, C. P.; Keating, J. P.; O’Connell, N., Random matrix theory and the derivative of the Riemann zeta function, Proc. R. Soc. London, Ser. A, 456, 2611-2627 (2000) · Zbl 0996.11052
[21] Mezzadri, F., Random matrix theory and the zeroes of ζ^′(s), J. Phys. A: Math. Gen., 36, 12, 2945-2962 (2003) · Zbl 1074.11048
[22] Conrey, J. B.; Rubinstein, M. O.; Snaith, N. C., Moments of the derivative of characteristic polynomials with an application to the Riemann zeta function, Commun. Math. Phys., 267, 3, 611-629 (2006) · Zbl 1210.11100
[23] Hughes, C. P., On the characteristic polynomial of a random unitary matrix and the Riemann zeta function (2001), University of Bristol
[24] Dehaye, P.-O., Joint moments of derivatives of characteristic polynomials, Alg. Number Theory, 2, 1, 31-68 (2008) · Zbl 1204.11143
[25] Riedtmann, H., A combinatorial approach to mixed ratios of characteristic polynomials (2018)
[26] Winn, B., Derivative moments for characteristic polynomials from the CUE, Commun. Math. Phys., 315, 531-562 (2012) · Zbl 1256.15021
[27] Conrey, J.; Forrester, P.; Snaith, N., Averages of ratios of characteristic polynomials for the compact classical groups, Int. Math. Res. Not., 2005, 7, 397-431 · Zbl 1156.11334
[28] Mason, A. M.; Snaith, N. C. (2018), AMS
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.