Large Positive and Negative Values of Hardy's $Z$-Function
| Autor: | Mahatab, Kamalakshya |
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| Rok vydání: | 2018 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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 }& \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*} Comment: 8 pages, some minor changes, To appear in Proceedings of the American Mathematical Society |
| Databáze: | arXiv |
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