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, Gál, and others. The paper aims at displaying and clarifying the conceptually different combinatorial arguments that show up in various parts of the proofs.

To appear in "Tribute to Victor Havin. 50 years with Hardy spaces", to be published as a volume in the series "Operator Theory: Advances and Applications", Birkhäuser Verlag