A generalization of Maynard's results on the Brun-Titchmarsh theorem to number fields

Abstract

Maynard proved that there exists an effectively computable constant $q_1$ such that if $q \geq q_1$, then $\frac{\log q}{\sqrt{q} \phi(q)} {\textup {Li}}(x) \ll \pi(x;q,m) \!<\! \frac{2}{\phi(q)} {\textup {Li}}(x)$ for $x \geq q^8$. In this paper, we will show the following. Let $\delta_1$ and $\delta_2$ be positive constants with $0< \delta_1, \delta_2 < 1$ and $\delta_1+\delta_2 > 1$. Assume that $L \neq {\mathbb Q}$ is a number field. Then there exist effectively computable constants $c_0$ and $d_1$ such that for $d_L \geq d_1$ and $x \geq \exp \left( 326 n_L^{\delta_1} \left(\log d_L\right)^{1+\delta_2}\right)$, we have $$\left| \pi_C(x) - \frac{|C|}{|G|} {\textup {Li}}(x) \right| \leq \left(1- c_0 \frac{\log d_L}{d_L^{7.072}} \right) \frac{|C|}{|G|} {\textup {Li}}(x).$$