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Proceedings of the American Mathematical Society

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Large positive and negative values of Hardy's $ Z$-function


Author: Kamalakshya Mahatab
Journal: Proc. Amer. Math. Soc. 147 (2019), 4161-4169
MSC (2010): Primary 11M06
DOI: https://doi.org/10.1090/proc/14483
Published electronically: June 27, 2019
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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$,

  $\displaystyle \quad \max _{T^{3/4}\leq t\leq T} Z(t) \gg \exp \left (\left (\fr... ...2}-\epsilon \right )\sqrt {\frac {\log T\log \log \log T}{\log \log T}}\right )$    
$\displaystyle \text {and}\quad$ $\displaystyle \quad \max _{T^{3/4}\leq t\leq T}- Z(t) \gg \exp \left (\left (\f... ...}-\epsilon \right )\sqrt {\frac {\log T\log \log \log T}{\log \log T}}\right ).$    


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Additional Information

Kamalakshya Mahatab
Affiliation: Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
Email: accessing.infinity@gmail.com, kamalakshya.mahatab@ntnu.no

DOI: https://doi.org/10.1090/proc/14483
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
Article copyright: © Copyright 2019 American Mathematical Society