Lower bounds for the maximum of the Riemann zeta function along vertical lines
By Christoph Aistleitner. Mathematische Annalen, 2014
72 citations recorded by the source corpus.
Let α∈(1/2,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (1/2,1)$$\end{document} be fixed. We prove that max0≤t≤T|ζ(α+it)|≥expcα(logT)1-α(loglogT)α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \max _{0 \le t \le T} |\zeta (\alpha +it)| \ge \exp \left( \frac{c_\alpha (\log T)^{1-\alpha }}{(\log \log T)^\alpha }\right) \end{aligned}$$\end{document}for all sufficiently large T, where we can choose cα=0.18(2α-1)1-α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_\alpha = 0.18 (2\alpha -1)^{1-\alpha }$$\end{document}. The same result has already been obtained by Montgomery, with a smaller value for cα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_\alpha $$\end{document}. However, our proof, which uses a modified version of Soundararajan’s “resonance method” together with ideas of Hilberdink, is completely different from Montgomery’s. This new proof also allows us to obtain lower bounds for the measure of those t∈[0,T]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t \in [0,T]$$\end{document} for which |ζ(α+it)|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\zeta (\alpha +it)|$$\end{document} is of the order mentioned above.
Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/26b290640343cdf8201ad745e9b80d821677f2a3
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Citations in the corpus
- Extreme values of the Riemann zeta function on the 1-line (2017, methodology): “The requirement for such a longer range is typical for applications of the “long resonator”, and appears in [1] and [3] as well.”
- Extreme values of the Riemann zeta function and its argument (2017, methodology): “We proceed as in [2] (following an idea from [1]) and let J be the set of integers j such that [ (1 + T−1) j , (1 + T−1) j+1 )⋂ M = ∅,”
- Sommes de Gál et applications (2018, unknown): “(ii) Notre approche diffère sensiblement de celle de [9] : à l’instar des méthodes développées dans [14] et [1], nous relions le carré du maximum à la somme de Gál complète S(M).”