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Computational estimation of the constant $ \beta (1)$ characterizing the order of $ \zeta (1+it)$


Author: Tadej Kotnik
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
Published electronically: January 24, 2008
MathSciNet review: 2398789
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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 $

$\displaystyle \frac{\zeta (2)}{2\beta (1)e^{\gamma }\left[ 1+o(1)\right] \log \... ... (1+it)\right\vert \leq 2\beta (1)e^{\gamma }\left[ 1+o(1) \right] \log \log t $

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.


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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

DOI: https://doi.org/10.1090/S0025-5718-08-02065-6
Keywords: Riemann's zeta function, line $\sigma =1$, constant $\beta (1)$
Received by editor(s): August 15, 2006
Received by editor(s) in revised form: April 26, 2007
Published electronically: January 24, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.