On the distribution of norms of prime ideals of a given class in arithmetic progressions (Q558743)

Let \(K=\mathbb Q(d)\), \(d < 0\), be an imaginary quadratic field of discriminant \(D\), \( \chi_{1}\) a character of \(K\), and \(R\) the ring of algebraic integers of the field \(K\). Let \(x\) be a positive rational integer, \(l\) and \(q\), \(1 \leq q \leq \log^{A_{1}} x\), \(A_{1}>0\), relatively prime rational integers, \( \mathcal {C}\) an ideal class of the ring \(R\), and \[ \pi(x,q,l,{ \mathcal C})=\sum_{{{P \in {\mathcal C}} \atop {N(P)\equiv l (\bmod\,q)}} \atop {N(P) \leq x}} 1 \] the number of prime ideals \(P \in { \mathcal C}\) whose norms lie in an arithmetic progression and do not exceed \(x\). The author proves that \[ \pi(x,q,,l, { \mathcal C})= \frac{1+ \chi(q;D, 0) \chi_{1}(l)}{h \varphi(q)} Li \text{ } x +O\left(x \exp(-c( \log x)^{1/ 20A_{1}})\right) \] as \(x \to \infty\), where \(h=h(D)\) is the class number of \(K\), \( \varphi (q)\) the Euler phi-function, \[ \chi(q; D,0)= \begin{cases} 1 & \text{if~} q \equiv 0(\bmod\, D) \cr 0 & \text{if~} q \not \equiv 0 (\bmod\, D),\cr \end{cases} \qquad Li \text{ } x = \int_{2}^{x} \frac{dx}{ \log x}, \] and \(c=c(A_{1})\) a positive constant.