Computational estimation of the constant $\beta (1)$ characterizing the order of $\zeta (1+it)$
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- by Tadej Kotnik PDF
- Math. Comp. 77 (2008), 1713-1723 Request permission
Abstract:
The paper describes a computational estimation of the constant $\beta (1)$ characterizing the bounds of $\left \vert \zeta (1+it)\right \vert$. It is known that as $t\rightarrow \infty$ \begin{equation*} \frac {\zeta (2)}{2\beta (1)e^{\gamma }\left [ 1+o(1)\right ] \log \log t}\leq \left \vert \zeta (1+it)\right \vert \leq 2\beta (1)e^{\gamma }\left [ 1+o(1) \right ] \log \log t \end{equation*} with $\beta (1)\geq \frac {1}{2}$, while the truth of the Riemann hypothesis would also imply that $\beta (1)\leq 1$. In the range $1<t\leq 10^{16}$, two sets of estimates of $\beta (1)$ are computed, one for increasingly small minima and another for increasingly large maxima of $\left \vert \zeta (1+it)\right \vert$. As $t$ increases, the estimates in the first set rapidly fall below $1$ and gradually reach values slightly below $0.70$, while the estimates in the second set rapidly exceed $\frac {1}{2}$ and gradually reach values slightly above $0.64$. The obtained numerical results are discussed and compared to the implications of recent theoretical work of Granville and Soundararajan.References
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Additional Information
- Tadej Kotnik
- Affiliation: Faculty of Electrical Engineering, University of Ljubljana, Tržaška 25, SI-1000 Ljubljana, Slovenia
- Email: tadej.kotnik@fe.uni-lj.si
- Received by editor(s): August 15, 2006
- Received by editor(s) in revised form: April 26, 2007
- Published electronically: January 24, 2008
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 77 (2008), 1713-1723
- MSC (2000): Primary 11M06, 11Y60; Secondary 11Y35, 65A05
- DOI: https://doi.org/10.1090/S0025-5718-08-02065-6
- MathSciNet review: 2398789