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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: slester@post.tau.ac.il
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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.

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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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References

REFERENCES

[1]Berndt, B. C.The number of zeros for ζ(k)(s). J. London Math. Soc. (2) 2 (1970), 577–580.CrossRefGoogle Scholar
[2]Burkill, J. C.The Lebesgue Integral (Cambridge University Press, Cambridge, 1971).Google Scholar
[3]Dueñez, E., Farmer, D. W., Froehlich, S., Hughes, C. P., Mezzadri, F. and Phan, T.Roots of the derivative of the Riemann zeta-function and of characteristic polynomials. Nonlinearity 23 (2010), no. 10, 2599–2621.Google Scholar
[4]Durrett, R.Probability: Theory and Examples, 4th edn. (Cambridge University Press, Cambridge, 2010).Google Scholar
[5]Gareav, M. Z. and Yildirim, C. Y.On small distances between ordinates of zeros of ζ(s) and ζ′(s). Int. Math. Res. Not. no. 21 (2007).Google Scholar
[6]Guo, C. R.On the zeros of ζ(s) and ζ′(s). J. Number Theory 54 (1995), 206–210.CrossRefGoogle Scholar
[7]Guo, C. R.On the zeros of the Derivative of the Riemann zeta function. Proc. London Math Soc. (3) 72 (1996), 28–62.Google Scholar
[8]Farmer, D. W. and Landau, H. Ki.–Siegel zeros and the zeros of the derivative of the Riemann zeta-function. Adv. Math. 230 (2012), no. 4–6, 2048–2064.Google Scholar
[9]Feng, S.A note on the zeros of the derivative of the Riemann zeta-function near the critical line. Acta Arith. 120 (2005), 59–68.Google Scholar
[10]Ki, H.The zeros of the derivative of the Riemann zeta-function near the critical line. Int. Math. Res. Not. no. 16 (2008).Google Scholar
[11]Lester, S. J. The distribution of the logarithmeic derivative of the Riemann zeta-function. Quart. J. Math., to appear.Google Scholar
[12]Levinson, N.More than one-third of the zeros of Riemann's zeta-function are on σ = 1/2. Adv. Math. 13 (1974), 383–436.Google Scholar
[13]Levinson, N. and Montgomery, H. L.Zeros of the derivatives of the Riemann zeta-function. Acta Math. 133 (1974), 49–65.Google Scholar
[14]Mezzadri, F.Random matrix theory and the zeros of ζ′(s). Random matrix theory. J. Phys. A 36 (2003), no. 12, 2945–2962.Google Scholar
[15]Radziwiłł, M. Gaps between zeros of ζ(s) and the distribution of zeros of ζ′(s). Adv. Math., to appear.Google Scholar
[16]Selberg, A.Contributions to the theory of the Riemann zeta-function. Arch. Math. Naturvid. 48 (1946), no. 589–155.Google Scholar
[17]Soundararajan, K.The horizontal distribution of zeros of ζ′(s). Duke Math. J. 91 (1998), 33–59.Google Scholar
[18]Speiser, A.Geometrisches zur Riemannschen zeta funktion. Math. Ann. 110 (1934), 514–521.Google Scholar
[19]Titchmarsh, E. C.The Theory of the Riemann Zeta-Function, 2nd edn, revised by Heath–Brown, D.R. (Oxford University Press, Oxford, 1986).Google Scholar
[20]Titchmarsh, E. C.The Theory of Functions, 2nd edn. (Oxford University Press, Oxford, 1976).Google Scholar
[21]Zhang, Y.On the zeros of ζ′(s) near the critical line. Duke Math. J. 110 (2001), 555–572.Google Scholar