Large greatest common divisor sums and extreme values of the Riemann zeta function

Andriy Bondarenko, Kristian Seip

Type: Article

Publication Date: 2017-01-26

Citations: 75

DOI: https://doi.org/10.1215/00127094-0000005x

Abstract

It is shown that the maximum of |ζ(1/2+it)| on the interval T1/2≤t≤T 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 1≤n1<⋯<nN∑k,ℓ=1Ngcd(nk,nℓ)nknℓ≪Nexp(AlogNlogloglogNloglogN), established in a recent paper of ours, cannot be taken smaller than 1.

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+	Decoupling, exponential sums and the Riemann zeta function	2016	Jean Bourgain
+ PDF Chat	Extreme values of zeta and L-functions	2008	K. Soundararajan
+	Extreme values of the Riemann zeta function	1977	Hugh L. Montgomery
+	On an Inequality Satisfied by the Zeta-Function of Riemann	1928	E. C. Titchmarsh
+	On the frequency of titchmarsh’s phenomenon for ζ(s)—III	1977	R. Balasubramanian K. Ramachandra
+ PDF Chat	GCD sums and complete sets of square-free numbers	2014	Andriy Bondarenko Kristian Seip
+ PDF Chat	An arithmetical mapping and applications to Ω-results for the Riemann zeta function	2009	Titus Hilberdink
+	Sums Involving Common Divisors	1986	Tony Dyer Glyn Harman
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+	On a Certain Integral in the Theory of Uniform Distribution	1951	J.F. Koksma