Siegel–Walfisz theorem

《Algebraic Number Theory》 · 第 381 页 · Section 11.3「The Prime Number Theorem for APs」, Corollary 11.19

Corollary 11.19 (The Siegel–Walfisz theorem) Let $c_1$ be the constant in Theorem 11.16, and suppose that $A$ is given, $A>0$. If $q\le(\log x)^A$ and $(a,q)=1$, then $\psi(x;q,a)=\frac{x}{\varphi(q)}+O_A\left(x\exp\left(-c_1\sqrt{\log x}\right)\right)$.