Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Large positive and negative values of Hardy’s $Z$-function
HTML articles powered by AMS MathViewer

by Kamalakshya Mahatab PDF
Proc. Amer. Math. Soc. 147 (2019), 4161-4169 Request permission

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
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11M06
  • Retrieve articles in all journals with MSC (2010): 11M06
Additional 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