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

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- 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

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## 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