On the zeta functions attached to ray classes
Let \(K/\mathbb{Q}\) be a finite extension (the author is concerned mainly with quadratic extensions) and let \({\mathfrak m}(\neq(0))\) be an ideal of the ring of integers of \(K\), \({\mathcal O}(K)\). Denote by \(I({\mathfrak m})\) the group generated by the ideals \({\mathfrak a}\) of \({\mathcal O}(K)\) prime to \({\mathfrak m}\) and by \(P({\mathfrak m})\) the subgroup of principal ideals. Let \[ P_{\mathfrak m}= \bigl\{(x) \in P({\mathfrak m}) \mid x\equiv 1\mod^\times {\mathfrak m} \bigr\} \] and \[ P_{{\mathfrak m}^+}= \bigl\{(x)\in P_{\mathfrak m} \mid\sigma_1 (x)>0,\dots, \sigma_r(n)> 0\bigr \}, \] where \(\sigma_1, \dots,\sigma_r\) are the \(\mathbb{Q}\)-monomorphisms \(K\to \mathbb{R}\). For a group \(\ell\) such that \(P_{{\mathfrak m}^+}\leq\ell\leq P({\mathfrak m})\) let \(H(\ell)= I({\mathfrak m})/ \ell\). For \(\text{Re} s>1\) the zeta function attached to the ray class \({\mathfrak a}\ell\) is defined by \[ \zeta(s,{\mathfrak a}\ell)= \sum_{n \ell ={\mathfrak a}\ell} {n{\mathfrak m}^s \over N{\mathfrak n}^s} \] where \(N\) denotes the norm. The main results of this paper are related to the problem of obtaining inequalities for \[ \int\bigl|\zeta(s,{\mathfrak a} \ell)\bigr |^2dt \] in the region \({1\over 2}\leq \sigma<1\), \(s=\sigma +it\). After proving some elementary results suggested by \textit{K. Prachar}'s [Primzahlverteilung. Springer-Verlag (1978; Zbl 0394.10001)] and after obtaining the functional equation and approximate functional equation (using ideas from \textit{K. Chandrasekharan} and \textit{R. Narasimhan} [Math. Ann. 152, 30-64 (1963; Zbl 0116.27001)], the author's main result is as follows. Consider the quadratic case and let \[ \zeta^*(s,{\mathfrak a}\ell) = \sum_{n \geq N{\mathfrak m}} {a({\mathfrak m}, {\mathfrak a},n)\over N {\mathfrak m}^s},\quad Z^* (\sigma)= N{\mathfrak m}^{2\sigma} \sum_{n\geq N{\mathfrak m}} {a({\mathfrak m}, {\mathfrak a}, n)^2 \over N{\mathfrak n}^{2\sigma}}, \] where \(a({\mathfrak m},{\mathfrak a}, n)\) denotes the number of ideals of \({\mathcal O}(K)\) such that \({\mathfrak b}\ell= {\mathfrak a}\ell\) and \(N{\mathfrak b}=n\). Let \(T\geq 8\) and \({1\over 2}<\sigma<{3\over 4}\), then \[ {1\over T} \int^T_8 \bigl|\zeta^*(s,{\mathfrak a}\ell) \bigr|^2 dt= Z^*(\sigma)+O_{k, \sigma} \bigl(q({\mathfrak m}, {\mathfrak a})\bigr)T^{{1\over 2}-\sigma}\log \varepsilon^2_m T, \] and there is a similar estimate for \(\int |\zeta^* ({1\over 2}+ it,{\mathfrak a}\ell)|^2 dt\). Here \(q({\mathfrak m},{\mathfrak n}) = \varepsilon^2_m{N {\mathfrak m}\over \rho({\mathfrak m}, {\mathfrak a})}\), where \(\varepsilon_m\) is the absolute value of a certain unit and \(\rho({\mathfrak m}, {\mathfrak a})\) denotes the minimum of \(N{\mathfrak m}\) and \(|N{\mathfrak b}- N{\mathfrak b}'|\) for \({\mathfrak b},{\mathfrak b}'\triangleleft{\mathcal O}(K)\), \({\mathfrak b},{\mathfrak b}'\in{\mathfrak a}\ell\). The methods rely on ideas to be found in \textit{H. L. Mongomery} [Topics in Multiplicative Number Theory, Lecture Notes in Mathematics 227, Springer-Verlag (1971; Zbl 0216.03501)].
zeta functions of algebraic number fields
scientific article; zbMATH DE number 858010