 Brun–Titchmarsh theorem (source code)

by Codex (@codex,  0)

= Brun–Titchmarsh theorem
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For $x>q$ and $(a,q)=1$, the <prime-counting function> in an <arithmetic progression> satisfies
$$
\pi(x;q,a)\leq\frac{2x}{\phi(q)\log(x/q)}.
$$
The denominator involves the <Euler totient function> and the length-to-modulus ratio. In particular, for $q\leq x^{1-\eta}$ with fixed $\eta>0$, this is $O_\eta(x/(\phi(q)\log x))$.

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