Discrete fundamental region and its asymptotics (Q1324849)

The discrete fundamental region means here the set of the numbers \(\omega_ j = (- b + i \sqrt D)/(2a)\) related to the reduced binary quadratic forms \(ax^ 2 + bxy + cy^ 2\) of the negative discriminant \(- D\); these numbers lie in the standard fundamental region of the full modular group. The number of the \(\omega_ j\) equals \(h(D)\), the class number of the field \(\mathbb{Q} (\sqrt{-D})\), and they occur in the formula \(\xi_ D (s) = \xi (2s) \sum^{h(D)}_{j = 1} E(\omega_ j,s)\), where \(\xi_ D (s) = (\sqrt D/2 \pi)^ s \Gamma(s) \zeta_ D(s)\) is related to the Dedekind zeta-function \(\zeta_ D(s)\) of the field \(\mathbb{Q}(\sqrt D)\), \(\xi(s) = \pi^{-s/2} \Gamma (s/2) \zeta(s)\), and \(E(z,s)\) denotes the Eisenstein-Maass series. Heuristically, the sum over \(\omega_ j\) may be approximated by an integral over a region in the fundamental domain. This heuristics is highly interesting, for it provides an approach to Riemann's hypothesis, as it has been shown by \textit{L. A. Takhtadjan} (L. A. Takhtadzhyan) and the author [Sov. Math., Dokl. 22, 136-140 (1980); translation from Dokl. Akad. Nauk SSSR 253, No. 4, 777- 781 (1980; Zbl 0488.10017)]. In the present paper, the error of such an approximation is considered in the mean square sense over \(D\), and this mean square is related to mean values of the form \(\sum_{D \leq N} L(w_ 1, \chi_ D) L(w_ 2, \chi_ D)\) involving quadratic Dirichlet \(L\)-functions of the fields in question. An asymptotic formula for this sum can be worked out though the author does not go into the complicated details.