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On the distribution of the zeros of the derivative of the Riemann zeta-function

Published online by Cambridge University Press:  29 September 2014

STEPHEN LESTER
STEPHEN LESTER*
Affiliation:
Department of Mathematics, University of Rochester, Rochester NY 14627, U.S.A. e-mail: [email protected]
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Abstract

We establish an asymptotic formula describing the horizontal distribution of the zeros of the derivative of the Riemann zeta-function. For ℜ(s) = σ satisfying (log T)−1/3+ε ⩽ (2σ − 1) ⩽ (log log T)−2, we show that the number of zeros of ζ′(s) with imaginary part between zero and T and real part larger than σ is asymptotic to T/(2π(σ−1/2)) as T → ∞. This agrees with a prediction from random matrix theory due to Mezzadri. Hence, for σ in this range the zeros of ζ′(s) are horizontally distributed like the zeros of the derivative of characteristic polynomials of random unitary matrices are radially distributed.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 157 , Issue 3 , November 2014 , pp. 425 - 442
Copyright
Copyright © Cambridge Philosophical Society 2014 

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