## 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

Zbl 1186.11049
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