Note on the resonance method for the Riemann zeta function
Journal article, Peer reviewed
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Original versionOperator Theory: Advances and Applications. 2018, 261 121-140. 10.1007/978-3-319-59078-3_6
We improve Montgomery’s Ω-results for |ζ(σ + it)| in the strip 1/2 σ 1 and give in particular lower bounds for the maximum of |ζ(σ+it)| on √ T ≤ t ≤ T that are uniform in σ. We give similar lower bounds for the maximum of |_n≤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 |ζ(1/2 + it)| on long intervals, as well as work of Soundararajan, G´al, and others. The paper aims at displaying and clarifying the conceptually different combinatorial arguments that show up in various parts of the proofs.