Prime number theorem in an arithmetic progression with an exceptional zero
= Prime number theorem in an arithmetic progression with an exceptional zero
Uniformly for $x\geq2$ and $(a,q)=1$,
$$
\psi(x;q,a)
=\frac{x}{\varphi(q)}
-\frac{\chi_1(a)x^\beta}{\beta\varphi(q)}
+O\left(
x(\log q)^2
\exp\left[-\frac{c\log x}{\log q+\sqrt{\log x}}\right]
\right),
$$
where the second term occurs only when a real character $\chi_1$ has an <exceptional zero> $\beta$.