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Discrepancy bounds for the distribution of the Riemann zeta-function and applications

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Abstract

We investigate the distribution of the Riemann zeta-function on the line Re(s) = σ. For ½ < σ ≤ 1 we obtain an upper bound on the discrepancy between the distribution of ζ (s) and that of its random model, improving results of Harman and Matsumoto. Additionally, we examine the distribution of the extreme values of ζ (s) inside of the critical strip, strengthening a previous result of the first author.

As an application of these results we obtain the first effective error term for the number of solutions to ζ (s) = a in a strip ½ < σ1 < σ2 < 1. Previously in the strip ½ < σ< 1 only an asymptotic estimate was available due to a result of Borchsenius and Jessen from 1948 and effective estimates were known only slightly to the left of the half-line, under the Riemann hypothesis (due to Selberg). In general our results are an improvement of the classical Bohr–Jessen framework and are also applicable to counting the zeros of the Epstein zeta-function

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Authors and Affiliations

  1. Institut Elie Cartan de Lorraine, Université de Lorraine, BP 70239, 54506, Vandoeuvre-Les-Nancy Cede, France

    Youness Lamzouri

  2. School Of Mathematical Sciences, Queen Mary University of London, 327 Mile End Road, London, E1 4NS, UK

    Stephen Lester

  3. Department of Mathematics, California Institute Of Technology, 1200 E California Blvd., Pasadena, CA, 91125, USA

    Maksym Radziwiłł

Authors
  1. Youness Lamzouri
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  2. Stephen Lester
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  3. Maksym Radziwiłł
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Corresponding author

Correspondence to Youness Lamzouri.

Additional information

The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no 320755.

The first author is supported in part by an NSERC Discovery grant.

The third author was partially supported by NSF grant DMS-1128155.

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Lamzouri, Y., Lester, S. & Radziwiłł, M. Discrepancy bounds for the distribution of the Riemann zeta-function and applications. JAMA 139, 453–494 (2019). https://doi.org/10.1007/s11854-019-0063-1

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  • DOI: https://doi.org/10.1007/s11854-019-0063-1

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