Note on the Resonance Method for the Riemann Zeta Function

  • Andriy Bondarenko
  • Kristian Seip
Part of the Operator Theory: Advances and Applications book series (OT, volume 261)


We improve Montgomery’s Ω-results for (σ + it)| in the strip 1/2 σ 1 and give in particular lower bounds for the maximum of (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 (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.


Resonance method Riemann zeta function Hardy spaces Dirichlet series 

Mathematics Subject Classification (2010)

Primary 11M06. Secondary 11C20 


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesNorwegian University of Science and TechnologyTrondheimNorway

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