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

Lamzouri, Youness and Lester, Stephen and Radziwiłł, Maksym (2019) Discrepancy bounds for the distribution of the Riemann zeta-function and applications. Journal d'Analyse Mathématique, 139 (2). pp. 453-494. ISSN 0021-7670. doi:10.1007/s11854-019-0063-1.

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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.

Item Type:Article
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Lester, Stephen0000-0003-1977-9205
Additional Information:© 2019 The Hebrew University of Jerusalem. Received 20 January 2017; First Online 05 November 2019. 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.
Funding AgencyGrant Number
European Research Council (ERC)320755
Natural Sciences and Engineering Research Council of Canada (NSERC)UNSPECIFIED
Issue or Number:2
Classification Code:2010 Mathematics Subject Classification: Primary 11M06
Record Number:CaltechAUTHORS:20180614-135438432
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Official Citation:Lamzouri, Y., Lester, S. & Radziwiłł, M. Discrepancy bounds for the distribution of the Riemann zeta-function and applications. JAMA 139, 453–494 (2019) doi:10.1007/s11854-019-0063-1
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:87119
Deposited By: George Porter
Deposited On:14 Jun 2018 21:04
Last Modified:15 Nov 2021 20:45

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