Moments of the derivative of the Riemann zeta-function and of characteristic polynomials

2005/08/19 by J. Brian Conrey, Conrey, J. Brian, Michael O. Rubinstein +3
Mathematics · Physics and Astronomy · #11M06 #15A52 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Number Theory (math.NT) #math-ph #math.MP #math.NT #msc:11M06 #msc:15A52

paper · pdf · doi:10.48550/arxiv.math/0508378

19 pages

arxiv created 2006/03/17 · arxiv updated 2009/12/01

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 zeta-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.

Related

• On moments of the derivative of CUE characteristic polynomials and the Riemann zeta function
• Complex moments of the derivative of the Riemann zeta function
• Moments of derivatives of the Riemann zeta function: Characteristic polynomials and the hybrid formula
• Joint moments of higher order derivatives of CUE characteristic polynomials I: asymptotic formulae
• Moments of the derivative of the characteristic polynomial of unitary matrices