Note on the Resonance Method for the Riemann Zeta Function
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.
KeywordsResonance method Riemann zeta function Hardy spaces Dirichlet series
Mathematics Subject Classification (2010)Primary 11M06. Secondary 11C20
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