A Bombieri-Vinogradov theorem for all number fields (Q2847130)

Let \(K/M\) be a Galois extension of number fields. For an unramified prime ideal \(\mathfrak{p}\) of \(M\), let \(\sigma_{\mathfrak{p}}\) be the corresponding Frobenius conjugacy class \(\sigma_{\mathfrak{p}}\). The goal of this paper is a Bombieri-Vinogradov theorem for prime ideals \(\mathfrak{p}\) for which the conjugacy class \(\sigma_{\mathfrak{p}}\) is specified. The paper extends work of [the first author and \textit{V. K. Murty}, in: Number theory, Proc. Conf., Montreal/Can. 1985, CMS Conf. Proc. 7, 243--272 (1987; Zbl 0619.10039)], which concerned the case \(M=\mathbb{Q}\).NEWLINENEWLINETo be more precise, let \(G=\)Gal\((K/M)\) and let \(C\) be a conjugacy class of \(G\). For coprime integers \(a\) and \(q\) with \(q\geq 1\) we write \(\pi(x,C,q,a)\) for the number of unramified prime ideals \(\mathfrak{p}\) of \(K\) of norm \(N(\mathfrak{p})\leq x\), and such that \(\sigma_{\mathfrak{p}}=C\) and \(N(\mathfrak{p})\equiv a\)(mod \(q)\). If NEWLINE\[NEWLINEK\cap\mathbb{Q}(\zeta_q)=\mathbb{Q},\;\;\; (*)NEWLINE\]NEWLINE then we have NEWLINE\[NEWLINE\pi(x,C,q,a)\sim\frac{|C|}{|G|\phi(q)}\pi(x),\;\;(x\rightarrow\infty),NEWLINE\]NEWLINE by the Chebotarev density theorem.NEWLINENEWLINEThe result of the present paper is then that NEWLINE\[NEWLINE{\sum_{q\leq x^{\theta}}^{}}^* \max_{y\leq x}\max_{(a,q)=1}\left| \pi(y,C,q,a)-\frac{|C|}{|G|\phi(q)}\pi(y)\right|\ll_{A,K,\theta} \frac{x}{(\log x)^A},NEWLINE\]NEWLINE where \(\Sigma^*\) indicates that the sum is restricted to \(q\) satisfying (*). Here \(\theta\) may be any fixed exponent such that NEWLINE\[NEWLINE\theta^{-1}>\max([E:\mathbb{Q}]-2\,,\,2),NEWLINE\]NEWLINE where \(E\) is the fixed field of \(H\), the largest abelian subgroup of \(G\) such that \(H\cap C\not=\emptyset\).NEWLINENEWLINEThe proof handles small \(q\) via a consideration of Siegel zeros for Artin \(L\)-functions. For the remaining range one uses a Vaughan type decomposition for Dirichlet polynomials involving the von Mangoldt function for \(K\), weighted by an appropriate group character. This is coupled with estimates of large sieve type for such characters.