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15 June 2017 Large greatest common divisor sums and extreme values of the Riemann zeta function
Andriy Bondarenko, Kristian Seip
Duke Math. J. 166(9): 1685-1701 (15 June 2017). DOI: 10.1215/00127094-0000005X

Abstract

It is shown that the maximum of |ζ(1/2+it)| on the interval T1/2tT is at least exp((1/2+o(1))logTlogloglogT/loglogT). Our proof uses Soundararajan’s resonance method and a certain large greatest common divisor sum. The method of proof shows that the absolute constant A in the inequality

sup 1n1<<nNk,=1Ngcd(nk,n)nknNexp(AlogNlogloglogNloglogN), established in a recent paper of ours, cannot be taken smaller than 1.

Citation

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Andriy Bondarenko. Kristian Seip. "Large greatest common divisor sums and extreme values of the Riemann zeta function." Duke Math. J. 166 (9) 1685 - 1701, 15 June 2017. https://doi.org/10.1215/00127094-0000005X

Information

Received: 5 August 2015; Revised: 23 July 2016; Published: 15 June 2017
First available in Project Euclid: 26 January 2017

zbMATH: 06745536
MathSciNet: MR3662441
Digital Object Identifier: 10.1215/00127094-0000005X

Subjects:
Primary: 11M06
Secondary: 11C20

Keywords: Extreme values , greatest common divisor sums , Riemann zeta function

Rights: Copyright © 2017 Duke University Press

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Vol.166 • No. 9 • 15 June 2017
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