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Large greatest common divisor sums and extreme values of the Riemann zeta function. (English) Zbl 1394.11063

In the paper, a new estimate for extreme values of the Riemann zeta function \(\zeta(s)\) on the half-line is established. Specifically, it is shown that the maximum of \(|\zeta(1/2+it)|\) on the interval \(T^1/2\leq t\leq T\) is at least \(\exp[(1/\sqrt 2+o(1))\sqrt{\log T\log\log\log T/\log\log T}]\). The proof of the authors uses Soundararajan’s resonance method and a certain large GCD sum.
The best lower estimate for extreme values of \(|\zeta(1/2+it)|\) known previously was obtained in 2008 by K. Soundararajan [Math. Ann. 342, No. 2, 467–486 (2008; Zbl 1186.11049)], who proved that \(|\zeta(1/2+it)|\geq\exp[(1+o(1))\sqrt{\log T/\log\log T}]\).

MSC:

11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11C20 Matrices, determinants in number theory

Citations:

Zbl 1186.11049
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