Note on the resonance method for the Riemann zeta function

Abstract

We improve Montgomery's $\Omega$-results for $|\zeta(\sigma+it)|$ in the strip $1/2<\sigma<1$ and give in particular lower bounds for the maximum of $|\zeta(\sigma+it)|$ on $\sqrt{T}\le t \le T$ that are uniform in $\sigma$. We give similar lower bounds for the maximum of $|\sum_{n\le x} n^{-1/2-it}|$ on intervals of length much larger than $x$. We rely on our recent work on lower bounds for maxima of $|\zeta(1/2+it)|$ on long intervals, as well as work of Soundararajan, G\'{a}l, and others. The paper aims at displaying and clarifying the conceptually different combinatorial arguments that show up in various parts of the proofs.