Hostname: page-component-848d4c4894-cjp7w Total loading time: 0 Render date: 2024-07-03T00:11:12.461Z Has data issue: false hasContentIssue false
  • English
  • Français

The mean square of the Dedekind zeta function in quadratic number fields

Published online by Cambridge University Press:  24 October 2008

Wolfgang Müller
Wolfgang Müller
Affiliation:
Institut für Statistik, Technische Universität Graz, Lessingstraβe 27, A-8010 Graz, Austria
Get access Rights & Permissions [Opens in a new window]

Extract

Let K be a quadratic number field with discriminant D. The aim of this paper is to study the mean square of the Dedekind zeta function ζK on the critical line, i.e.

It was proved by Chandrasekharan and Narasimhan[1] that (1) is at most of order O(T(log T)2). As they noted at the end of their paper, it ‘would seem likely’ that (1) behaves asymptotically like a2T(log T)2, with some constant a2 depending on K. Applying a general mean value theorem for Dirichlet polynomials, one can actually prove

This may be done in just the same way as this general mean value theorem can be used to prove Ingham's classical result on the fourth power moment of the Riemann zeta function (cf. [3], chapter 5). In 1979 Heath-Brown [2] improved substantially on Ingham's result. Adapting his method to the above situation a much better result than (2) can be obtained. The following Theorem deals with a slightly more general situation. Note that ζK(s) = ζ(s)L(s, XD) where XD is a real primitive Dirichlet character modulo |D|. There is no additional difficulty in allowing x to be complex.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 106 , Issue 3 , November 1989 , pp. 403 - 417
Copyright
Copyright © Cambridge Philosophical Society 1989

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

REFERENCES

[1]Chandrasekharan, K. and Narasimhan, R.. The approximate functional equation for a class of zeta functions. Math. Ann. 152 (1963), 30–64.CrossRefGoogle Scholar
[2]Heath-Brown, D. R.. The fourth power moment of the Riemann zeta function. Proc. London Math. Soc. (3) 38 (1979), 385–422.CrossRefGoogle Scholar
[3]Ivić, A.. The Riemann Zeta Function (Wiley, 1985).Google Scholar
[4]Kuipers, L. and Niederreiter, H.. Uniform Distribution of Sequences (Wiley, 1974).Google Scholar
[5]McCarthy, P. J.. Introduction to Arithmetical Functions (Springer-Verlag, 1986).CrossRefGoogle Scholar
[6]Müller, W.. On the asymptotic behaviour of the ideal counting function in quadratic number fields. Monatsh. Math. (to appear).Google Scholar
[7]Montgomery, H. L.. Topics in Multiplicative Number Theory (Springer-Verlag, 1971).CrossRefGoogle Scholar
[8]Prachar, K.. Primzahlverteilung (Springer-Verlag, 1957).Google Scholar
[9]Rademacher, H.. On the Phragmén-Lindelöf theorem and some applications. Math. Z. 72 (1959), 192–204.CrossRefGoogle Scholar
[10]Titchmarsh, E. C.. The Theory of the Riemann Zeta Function (second edition, revised by Heath-Brown, D. R.) (Clarendon Press, 1986).Google Scholar