Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-23T04:56:15.531Z Has data issue: false hasContentIssue false
  • English
  • Français

The large values of the Riemann Zeta-function

Part of: Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  26 February 2010

Kai-Man Tsang
Kai-Man Tsang
Affiliation:
Department of Mathematics, University of Hong Kong, Pokfulam Road, Hong Kong.
Get access

Abstract

Let |θ| < π/2 and . By refining Selberg's method, we study the large values of as t → ∞ For σ close to ½ we obtain Ω+ estimates that are as good as those obtained previously on the Riemann Hypothesis. In particular, we show that

and

Our results supplement those of Montgomery which are good when σ > ½ is fixed.

MSC classification

Secondary: 11M06: $zeta (s)$ and $L(s, chi)$
Type
Research Article
Information
Mathematika , Volume 40 , Issue 2 , December 1993 , pp. 203 - 214
Copyright
Copyright © University College London 1993

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Bateman, H.. Tables of Integral Transforms (McGraw-Hill, 1954).Google Scholar
2.Balasubramanian, R. and Ramachandra, K.. On the frequency of Titchmarsh's phenomenon for ζ(s)-III. Proc. Indian Acad, Sci., 83A (1977), 341–351.Google Scholar
3.Levinson, N.. Ω-theorems for the Riemann zeta-function. Acta Arith., 20 (1972), 317–330.Google Scholar
4.Montgomery, H.. Extreme values of the Riemann zeta function. Comment. Math. Helv., 52 (1977), 511–518.Google Scholar
5.Selberg, A.. Contributions to the theory of the Riemann zeta-function. Arch, for Math, og Naturv. B, 48 (1946), no. 5.Google Scholar
6.Titchmarsh, E. C.. On an inequality satisfied by the zeta-function of Riemann. Proc. London Math. Soc., 28 (1928), 70–80.Google Scholar
7.Titchmarsh, E. C.. The Theory of the Riemann Zeta-function, 2nd ed. Revised by Heath-Brown, D. R. (Oxford University Press, Oxford, 1988).Google Scholar
8.Tsang, K.. Some Ω-theorems for the Riemann zeta-function. Acta Arith., 46 (1986), 369–395.Google Scholar