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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Large positive and negative values of Hardy’s $Z$-function
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by Kamalakshya Mahatab
Proc. Amer. Math. Soc. 147 (2019), 4161-4169
DOI: https://doi.org/10.1090/proc/14483
Published electronically: June 27, 2019

Abstract:

Let $Z(t):=\zeta \left (\frac {1}{2}+it\right )\chi ^{-\frac {1}{2}}\left (\frac {1}{2}+it\right )$ be Hardy’s function, where the Riemann zeta function $\zeta (s)$ has the functional equation $\zeta (s)=\chi (s)\zeta (1-s)$. We prove that for any $\epsilon >0$, \begin{align*} &\quad \max _{T^{3/4}\leq t\leq T} Z(t) \gg \exp \left (\left (\frac {1}{2}-\epsilon \right )\sqrt {\frac {\log T\log \log \log T}{\log \log T}}\right )\\ \text {and}\quad &\quad \max _{T^{3/4}\leq t\leq T}- Z(t) \gg \exp \left (\left (\frac {1}{2}-\epsilon \right )\sqrt {\frac {\log T\log \log \log T}{\log \log T}}\right ). \end{align*}
References
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Bibliographic Information
  • Kamalakshya Mahatab
  • Affiliation: Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
  • MR Author ID: 1037301
  • Email: accessing.infinity@gmail.com, kamalakshya.mahatab@ntnu.no
  • Received by editor(s): October 4, 2018
  • Published electronically: June 27, 2019
  • Additional Notes: The author was supported by Grant 227768 of the Research Council of Norway
  • Communicated by: Amanda Folsom
  • © Copyright 2019 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 147 (2019), 4161-4169
  • MSC (2010): Primary 11M06
  • DOI: https://doi.org/10.1090/proc/14483
  • MathSciNet review: 4002533