Abstract
We investigate the moments of the derivative, on the unit circle, of characteristic polynomials of random unitary matrices and use this to formulate a conjecture for the moments of the derivative of the Riemann ζ function on the critical line. We do the same for the analogue of Hardy’s Z-function, the characteristic polynomial multiplied by a suitable factor to make it real on the unit circle. Our formulae are expressed in terms of a determinant of a matrix whose entries involve the I-Bessel function and, alternately, by a combinatorial sum.
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References
Brezin E., Hikami S. (2000). Characteristic polynomials of random matrices at edge singularities. Phys. Rev. E 62(3):3558–3567
Conrey J.B., Farmer D.W., Keating J.P., Rubinstein M.O., Snaith N.C. (2005). Integral moments of L-functions. Proc. London Math. Soc. 91:33–104
Conrey J.B., Farmer D.W., Keating J.P., Rubinstein M.O., Snaith N.C. (2003). Autocorrelation of random matrix polynomials. Commun. Math. Phys. 237(3):365–395
Conrey, J.B., Ghosh, A.: Zeros of derivatives of the Riemann zeta-function near the critical line. Analytic number theory (Allerton Park, IL, 1989), Progr. Math. 85, Boston, MA: Birkhäuser Boston, 1990, pp. 95–110
Forrester, P.J., Witte, N.S.: Boundary conditions associated with the Painlevé III′ and V evaluations of some random matrix averages. http://arXiv.org/list/math.CA/0512142
Gonek, S.M., Hughes, C.P., Keating, J.P.: hybrid Euler-Hadamard product formula for the Riemann zeta function. http://arXiv.org/list/math.NT/0511182, 2005
Hughes, C.P.: On the characteristic polynomial of a random unitary matrix and the Riemann zeta function. PhD thesis, University of Bristol, 2001
Keating J.P., Snaith N.C. (2000). Random matrix theory and ζ(1/2 + it). Commun. Math. Phys. 214(1):57–89
Levinson N. (1974). More than one third of zeros of Riemann’s zeta-function are on σ = 1/2. Adv. Math. 13:383–436
Levinson N., Montgomery H.L. (1974). Zeros of the derivatives of the Riemann zeta-function. Acta Math. 133:49–65
Mezzadri F. (2003). Random matrix theory and the zeros of ζ′(s). J. Phys. A 36(12):2945–2962
Soundararajan K. (1998). The horizontal distribution of zeros of ζ′(s). Duke Math. J. 91(1):33–59
Weyl H. (1946). The Classical Compact Groups. Princeton University Press, Princeton, NJ
Zhang Y. (2001). On the zeros of ζ′(s) near the critical line. Duke Math. J. 110(3):555–572
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Communicated by P. Sarnak
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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, 611–629 (2006). https://doi.org/10.1007/s00220-006-0090-5
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DOI: https://doi.org/10.1007/s00220-006-0090-5