caa a22 4500 386386161 CHVBK 20180307112058.0 cr unu---uuuuu 161130e198911 xx s 000 0 eng 10.1017/S0305004100068134 doi S0305004100068134 pii (NATIONALLICENCE)cambridge-10.1017/S0305004100068134 Müller Wolfgang Institut für Statistik, Technische Universität Graz, Lessingstraβe 27, A-8010 Graz, Austria The mean square of the Dedekind zeta function in quadratic number fields [Elektronische Daten] [Wolfgang Müller] 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. Copyright © Cambridge Philosophical Society 1989 Mathematical Proceedings of the Cambridge Philosophical Society Cambridge University Press 106/3(1989-11), 403-417 0305-0041 106:3<403 1989 106 PSP https://doi.org/10.1017/S0305004100068134 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1017/S0305004100068134 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Müller Wolfgang Institut für Statistik, Technische Universität Graz, Lessingstraβe 27, A-8010 Graz, Austria NATIONALLICENCE 773 0- Mathematical Proceedings of the Cambridge Philosophical Society Cambridge University Press 106/3(1989-11), 403-417 0305-0041 106:3<403 1989 106 PSP CC0 http://creativecommons.org/publicdomain/zero/1.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-cambridge