Large GCD sums and extreme values of the Riemann zeta function

Abstract

It is shown that the maximum of $|\zeta (1/2+it)|$ on the interval $T^{1/2}\le t\le T$ is at least $\exp \left((1/\sqrt{2}+o(1))\sqrt{\log T\log \log \log T/\log \log T}\right)$. Our proof uses Soundararajan's resonance method and a certain large GCD sum. The method of proof shows that the absolute constant $A$ in the inequality \[ \sup_{1\le n_1<\cdots < n_N} \sum_{k,{\ell}=1}^N\frac{\gcd(n_k,n_{\ell})}{\sqrt{n_k n_{\ell}}} \ll N \exp\left(A\sqrt{\frac{\log N \log\log\log N}{\log\log N}}\right), \] established in a recent paper of ours, cannot be taken smaller than $1$.

This is the final version of this paper, to appear in Duke Math. J

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