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Real moments of the logarithmic derivative of characteristic polynomials in random matrix ensembles (2304.02153v4)

Published 4 Apr 2023 in math-ph, math.MP, and math.NT

Abstract: We prove asymptotics for real moments of the logarithmic derivative of characteristic polynomials evaluated at $1-\frac{a}{N}$ in unitary, even orthogonal, and symplectic ensembles, where $a>0$ and $a=o(1)$ as the size $N$ of the matrix goes to infinity. Previously, such asymptotics were known only for integer moments (in the unitary ensemble by the work of Bailey, Bettin, Blower, Conrey, Prokhorov, Rubinstein and Snaith, and in orthogonal and symplectic ensembles by the work of Alvarez and Snaith), except that in the odd orthogonal ensemble real moments asymptotics were obtained by Alvarez, Bousseyroux and Snaith. Our proof is new and does not make use of the aforementioned integer moments results, and is different from the method in the work of Alvarez et al for the odd orthogonal ensemble.

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References (31)
  1. Asymptotics of non-integer moments of the logarithmic derivative of characteristic polynomials over S⁢O⁢(2⁢N+1)𝑆𝑂2𝑁1SO(2N+1)italic_S italic_O ( 2 italic_N + 1 ). arXiv preprint arXiv:2303.07813, 2023.
  2. E. Alvarez and N. C. Snaith. Moments of the logarithmic derivative of characteristic polynomials from S⁢O⁢(N)𝑆𝑂𝑁SO(N)italic_S italic_O ( italic_N ) and U⁢S⁢p⁢(2⁢N)𝑈𝑆𝑝2𝑁USp(2N)italic_U italic_S italic_p ( 2 italic_N ). J. Math. Phys., 61(10):103506, 32, 2020.
  3. Siegfred Alan C. Baluyot. On the pair correlation conjecture and the alternative hypothesis. J. Number Theory, 169:183–226, 2016.
  4. Mixed moments of characteristic polynomials of random unitary matrices. J. Math. Phys., 60(8):083509, 26, 2019.
  5. Autocorrelation of ratios of L𝐿Litalic_L-functions. Commun. Number Theory Phys., 2(3):593–636, 2008.
  6. Brian Conrey. Notes on eigenvalue distributions for the classical compact groups. In Recent perspectives in random matrix theory and number theory, volume 322 of London Math. Soc. Lecture Note Ser., pages 111–145. Cambridge Univ. Press, Cambridge, 2005.
  7. Correlations of eigenvalues and Riemann zeros. Commun. Number Theory Phys., 2(3):477–536, 2008.
  8. Mean values of ζ′/ζ⁢(s)superscript𝜁′𝜁𝑠\zeta^{\prime}/\zeta(s)italic_ζ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT / italic_ζ ( italic_s ), correlations of zeros and the distribution of almost primes. Q. J. Math., 64(4):1057–1089, 2013.
  9. Landau-Siegel zeros and zeros of the derivative of the Riemann zeta function. Adv. Math., 230(4-6):2048–2064, 2012.
  10. Fan Ge. The distribution of zeros of ζ′⁢(s)superscript𝜁′𝑠\zeta^{\prime}(s)italic_ζ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_s ) and gaps between zeros of ζ⁢(s)𝜁𝑠\zeta(s)italic_ζ ( italic_s ). Adv. Math., 320:574–594, 2017.
  11. Fan Ge. The number of zeros of ζ′⁢(s)superscript𝜁′𝑠\zeta^{\prime}(s)italic_ζ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_s ). Int. Math. Res. Not. IMRN, (5):1578–1588, 2017.
  12. Fan Ge. Mean values of the logarithmic derivative of the riemann zeta-function near the critical line. Mathematika, 69(2):562–572, 2023.
  13. Fan Ge. On the logarithmic derivative of characteristic polynomials for random unitary matrices. Bull. Lond. Math. Soc., 56(3):1114–1128, 2024.
  14. Mean values of the logarithmic derivative of the Riemann zeta-function with applications to primes in short intervals. J. Reine Angew. Math., 537:105–126, 2001.
  15. Pair correlation of zeros and primes in short intervals. In Analytic number theory and Diophantine problems (Stillwater, OK, 1984), volume 70 of Progr. Math., pages 183–203. Birkhäuser Boston, Boston, MA, 1987.
  16. D. A. Goldston. On the pair correlation conjecture for zeros of the Riemann zeta-function. J. Reine Angew. Math., 385:24–40, 1988.
  17. Charng Rang Guo. On the zeros of the derivative of the Riemann zeta function. Proc. London Math. Soc. (3), 72(1):28–62, 1996.
  18. On small distances between ordinates of zeros of ζ⁢(s)𝜁𝑠\zeta(s)italic_ζ ( italic_s ) and ζ′⁢(s)superscript𝜁′𝑠\zeta^{\prime}(s)italic_ζ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_s ). Int. Math. Res. Not. IMRN, (21):Art. ID rnm091, 14, 2007.
  19. Haseo Ki. The zeros of the derivative of the Riemann zeta function near the critical line. Int. Math. Res. Not. IMRN, (16):Art. ID rnn064, 23, 2008.
  20. Random matrices, Frobenius eigenvalues, and monodromy, volume 45 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 1999.
  21. Random matrix theory and ζ⁢(1/2+i⁢t)𝜁12𝑖𝑡\zeta(1/2+it)italic_ζ ( 1 / 2 + italic_i italic_t ). Comm. Math. Phys., 214(1):57–89, 2000.
  22. S. J. Lester. The distribution of the logarithmic derivative of the Riemann zeta-function. Q. J. Math., 65(4):1319–1344, 2014.
  23. Zeros of the derivatives of the Riemann zeta-function. Acta Math., 133:49–65, 1974.
  24. H. L. Montgomery. The pair correlation of zeros of the zeta function. In Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972), pages 181–193, 1973.
  25. Orthogonal and symplectic n𝑛nitalic_n-level densities. Mem. Amer. Math. Soc., 251(1194):v+93, 2018.
  26. Hilbert’s inequality. J. London Math. Soc. (2), 8:73–82, 1974.
  27. Maksym Radziwiłł. Gaps between zeros of ζ⁢(s)𝜁𝑠\zeta(s)italic_ζ ( italic_s ) and the distribution of zeros of ζ′⁢(s)superscript𝜁′𝑠\zeta^{\prime}(s)italic_ζ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_s ). Adv. Math., 257:6–24, 2014.
  28. Atle Selberg. On the normal density of primes in small intervals, and the difference between consecutive primes. Arch. Math. Naturvid., 47(6):87–105, 1943.
  29. Atle Selberg. Contributions to the theory of the Riemann zeta-function. Arch. Math. Naturvid., 48(5):89–155, 1946.
  30. K. Soundararajan. The horizontal distribution of zeros of ζ′⁢(s)superscript𝜁′𝑠\zeta^{\prime}(s)italic_ζ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_s ). Duke Math. J., 91(1):33–59, 1998.
  31. Yitang Zhang. On the zeros of ζ′⁢(s)superscript𝜁′𝑠\zeta^{\prime}(s)italic_ζ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_s ) near the critical line. Duke Math. J., 110(3):555–572, 2001.
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  1. Fan Ge
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