# 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.

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

by Kamalakshya Mahatab
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*}
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