Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. H. Bohr and E. Landau, Über das Verhalten von $ \zeta (s)$ und $ \zeta _{\Re }(s)$ in der Nähe der Geraden $ \sigma =1$, Gött. Nachr. (1910) 303-330.
  • 2. J. E. Littlewood, On the Riemann zeta-function, Proc. Lond. Math. Soc. 24 (1925) 175-201.
  • 3. J. E. Littlewood, On the function $ 1/\zeta (1+ti)$, Proc. Lond. Math. Soc. 27 (1928) 349-357.
  • 4. E. C. Titchmarsh, On an inequality satisfied by the zeta-function of Riemann, Proc. Lond. Math. Soc. 28 (1928) 70-80.
  • 5. E. C. Titchmarsh, On the function $ 1/\zeta (1+ti)$, Quart. J. Math. Oxford 4 (1933) 64-70.
  • 6. S. Chowla, Improvement of a theorem of Linnik and Walfisz, Proc. Lond. Math. Soc. 50 (1948) 423-429. MR 0027302 (10:285d)
  • 7. N. Levinson, $ \Omega $-theorems for the Riemann zeta-function, Acta Arith. 20 (1972) 319-332. MR 0306135 (46:5262)
  • 8. A. Granville and K. Soundararajan, Extreme values of $ \left\vert \zeta (1+it)\right\vert $, ArXiv preprint math.NT/0501232 v1, http://arxiv.org/pdf/math/0501232.
  • 9. G. H. Hardy and J. E. Littlewood, The approximate functional equations for $ \zeta (s)$ and $ \zeta ^{2}(s)$, Proc. Lond. Math. Soc. 29 (1929) 81-97.
  • 10. E. C. Titchmarsh and D. R. Heath-Brown, The Theory of the Riemann Zeta-function, 2nd ed., Oxford University Press, 1986. MR 882550 (88c:11049)
  • 11. T. Kotnik, Computational estimation of the order of $ \zeta (\frac{1}{2}+it)$, Math. Comp. 73 (2004) 949-956. MR 2031417 (2004i:11098)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 11M06, 11Y60, 11Y35, 65A05

Retrieve articles in all journals with MSC (2000): 11M06, 11Y60, 11Y35, 65A05


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.

American Mathematical Society