Duke Mathematical Journal

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

Andriy Bondarenko and Kristian Seip

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Abstract

It is shown that the maximum of $\vert \zeta(1/2+it)\vert $ on the interval $T^{1/2}\le t\le T$ is at least $\exp ((1/\sqrt{2}+o(1))\sqrt{\log T\log\log\log T/\log\log T})$. 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

\[\sup_{1\le n_{1}\lt \cdots\lt n_{N}}\sum_{k,{\ell}=1}^{N}\frac{\operatorname{gcd}(n_{k},n_{\ell})}{\sqrt{n_{k}n_{\ell}}}\ll N\exp (A\sqrt{\frac{\log N\log\log\logN}{\log\log N}}),\] established in a recent paper of ours, cannot be taken smaller than $1$.

Article information

Source
Duke Math. J. Volume 166, Number 9 (2017), 1685-1701.

Dates
Received: 5 August 2015
Revised: 23 July 2016
First available in Project Euclid: 26 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1485400054

Digital Object Identifier
doi:10.1215/00127094-0000005X

Subjects
Primary: 11M06: $\zeta (s)$ and $L(s, \chi)$
Secondary: 11C20: Matrices, determinants [See also 15B36]

Keywords
greatest common divisor sums Riemann zeta function extreme values

Citation

Bondarenko, Andriy; Seip, Kristian. Large greatest common divisor sums and extreme values of the Riemann zeta function. Duke Math. J. 166 (2017), no. 9, 1685--1701. doi:10.1215/00127094-0000005X. https://projecteuclid.org/euclid.dmj/1485400054.


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