Large GCD sums and extreme values of the Riemann zeta function

TL;DR

This paper establishes a new lower bound for the maximum size of the Riemann zeta function on the critical line, using resonance methods and large GCD sums, revealing the growth rate of extreme values.

Contribution

It introduces a novel lower bound for the maximum of |(1/2+it)| and connects it with bounds on large GCD sums, advancing understanding of zeta function extremal behavior.

Findings

Lower bound for |(1/2+it)| growth rate.

Demonstrates the sharpness of the constant in GCD sum bounds.

Uses resonance method and large GCD sums in proof.

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 $1$.