Algebraic Number Theory
《Algebraic Number Theory》 · 第 90 页 · Section 3.5「Primes in arithmetic progressions」, Theorem 3.9
Theorem 3.9 (Brun–Titchmarsh) Let $a$ and $q$ be integers with $(a,q)=1$, and let $x$ and $y$ be real numbers with $x\ge0$ and $y\ge2q$. Then $\pi(x+y;q,a)-\pi(x;q,a)\le\frac{2y}{\varphi(q)\log(y/q)}\left(1+O\left(\frac1{\log(y/q)}\right)\right)$.