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• W2401780273 abstract "We show in this paper that after proper scalings, the characteristic polynomial of a random unitary matrix converges almost surely to a random analytic function whose zeros, which are on the real line, form a determinantal point process with sine kernel. Our scaling is performed at the so-called ”microscopic” level, that is we consider the characteristic polynomial at points which are of order 1/n distant. We draw several consequences from our result. On the random matrix theory side, we obtain the limiting distribution for ratios of characteristic polynomials where the points are evaluated at points of the form exp(2iπα/n). We also give an explicit expression for the (dependence) relation between two different values of the characteristic polynomial on the microscopic scale. On the number theory side, inspired by the Keating-Snaith philosophy, we conjecture some new limit theorems for the Riemann zeta function at the stochastic process level as well as some alternative approach to the conjecture by Goldston, Montgomery and Gonek for the moments of the logarithmic derivative of the Riemann zeta function. We prove our main random matrix theory result in the framework of virtual isometries to circumvent the fact that the rescaled characteristic polynomial does not even have a moment of order one, hence making the classical techniques of random matrix theory difficult to apply. ∗reda.chhaibi@math.uzh.ch †joseph.najnudel@math.univ-toulouse.fr ‡ashkan.nikeghbali@math.uzh.ch" @default.
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• W2401780273 date "2014-10-06" @default.
• W2401780273 modified "2023-10-01" @default.
• W2401780273 title "The Circular Unitary Ensemble and the Riemann zeta function: the microscopic landscape" @default.
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• W2401780273 hasPublicationYear "2014" @default.
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