# Reference for Brun-Titchmarsh inequality

Does anyone know a proof of the Brun-Titchmarsh inequality in the following form starting from the large sieve inequality?

Brun-Titchmarsh inequality: Let $\pi(x;q,a) = |\{p \text{ prime}: p\equiv a \pmod q , p\le x\}|$, $(a,q) = 1$. Then $$\pi(x;q,a) \ll \frac{x}{\phi(q)} \frac{1}{\log(\frac xq)} \quad \text{for }q<x.$$ with an absolute implied constant.

By the large sieve inequality I mean

Large sieve inequality: Fix $M,N\ge 1$, $Q\ge 1$. For primes $p\le Q$, let $\Omega_p\subset \mathbb Z/p\mathbb Z$ be given. Let $\omega = |\Omega_p|$, and $S=\{M\le n\le M+N: n\pmod p\notin \Omega_p \text{ for }p\le Q\}$. Then $$|S|\le \frac{N+Q^2}{H},$$ where $H=\sum_{\substack{q\le Q, \\ q\text{ squarefree}}} \prod_{p\mid q} \frac{\omega_p}{p-\omega_p}$.

If you can prove it or know a reference, I'd be grateful. Thanks for your help!

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