Journal of the
Korean Mathematical Society JKMS

Maynard proved that there exists an effectively computable constant $q_1$ such that
if $q \geq q_1$, then $\pi(x;q,a) < \frac{2 { {Li}}(x)}{\phi(q)}$
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 exists an effectively computable constant $d_1$ 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
$\pi_C(x) < 2 \frac{|C|}{|G|} {\ {Li}}(x)$.

Keywords: the Chebotarev density theorem, the Deuring-Heilbronn phenomenon, the Brun-Titchmarsh theorem

MSC numbers: 11R44, 11R42, 11M41, 11R45