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

초록

Maynard proved that there exists an effectively computable constant $q_1$ such that if $q\ge q_1$, then $\frac{\log q}{\sqrt{q}\varphi (q)}\text{Li}(x)\ll \pi (x;q,m)<\frac{2}{\varphi (q)}\text{Li}(x)$ for $x\ge q^8$. 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 $L\ne \mathbb{Q}$ is a number field. Then there exist effectively computable constants $c_0$ and $d_1$ such that for $d_L\ge d_1$ and $x\ge \exp \left(326n_L^{{\delta }_1}{\left(\log d_L\right)}^{1+{\delta }_2}\right)$, we have $\left|{\pi }_C(x)-\frac{|C|}{|G|}\text{Li}(x)\right|\le \left(1-c_0\frac{\log d_L}{d_L^{7.072}}\right)\frac{|C|}{|G|}\text{Li}(x).$

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