For $q,a$ relatively prime, let $\pi(x,q,a)$ denote the number of primes less than $x$ which are congruent to $a$ modulo $q$. The [Brun-Titchmarsh theorem](https://en.wikipedia.org/wiki/Brun%E2%80%93Titchmarsh_theorem) states that $$\pi(x,q,a)\leq \frac{(2+o(1))x}{\phi(q)\log(x/q)}$$ for all $q<x$. Letting $\theta=\frac{\log q}{\log x}$, the Brun-Titchmarsh theorem may be rewritten as $$\pi(x,q,a)\leq(C+o(1)) \frac{x}{\phi(q)\log x},$$ where $C=\frac{2}{1-\theta}$. There have been some improvements to this theorem, depending on the range of $\theta$. The optimal results based on the range are:


* $\begin{array}{ccc}
\frac{2}{3}\leq\theta & \Rightarrow & C=\frac{2-((1-\theta)/4)^{6}}{1-\theta}\end{array}. $ ([Friedlander and Iwaniec](https://web.archive.org/web/20160809134411/http://ebooks.cambridge.org/chapter.jsf?bid=CBO9780511666179&cid=CBO9780511666179A012))
* $\begin{array}{ccc}
\frac{9}{20}\leq\theta\leq\frac{2}{3} & \Rightarrow & C=\frac{8}{6-7\theta}\end{array}. $ ([Iwaniec](https://projecteuclid.org/journals/journal-of-the-mathematical-society-of-japan/volume-34/issue-1/On-the-Brun-Titchmarsh-theorem/10.2969/jmsj/03410095.full))
* $\begin{array}{ccc}
\frac{1}{8}\leq\theta\leq\frac{9}{20} & \Rightarrow & C=\frac{16}{8-3\theta}\end{array}. $ ([Motohashi](https://projecteuclid.org/journals/journal-of-the-mathematical-society-of-japan/volume-26/issue-2/On-some-improvements-of-the-Brun-Titchmarsh-theorem/10.2969/jmsj/02620306.full))
* $\begin{array}{ccc}
\theta\leq\frac{1}{8} & \Rightarrow & C=2\end{array}.$ ([Maynard](https://arxiv.org/abs/1201.1777))


(For $\theta$ very close to $1$, [Bourgain and Garaev](https://arxiv.org/abs/1309.1124) recently showed that we may take $C=\frac{2-c\_{1}(1-\theta)^{2}}{1-\theta}$ for an absolute constant $c\_1$.)



> 
> **Question:** Can the Brun-Titchmarsh theorem be improved further if the modulus $q$ is $y$ smooth for $y=x^\delta$ a small power of $x$? That is, if $q$ is composed of prime factors less that $x^\delta$, can we show that $$\pi(x,q,a)\leq (2+o(1)) \frac{x}{\phi(q)\log x}$$ for all $q$ up to $x$?
> 
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Can the Brun-Titchmarsh theorem be improved when the modulus is smooth?
(Credit goes also to James Maynard for this answer.) If you assume that $x\ge q^{12/5+\epsilon}$ for some fixed $\epsilon>0$ and you take $q$ to be $x^\theta$-smooth, then you do get that 
$$
\pi(x;q,a)\le (2+\delta) \frac{x}{\phi(q)\log x},
$$
with $\delta$ tending to 0 as $\theta\to0$ and $x\to\infty$. This follows by Theorem 10 in this paper of Chang <http://math.ucr.edu/~mcc/paper/143%20Char060113.pdf> and the zero-density estimates of Jutila (On Linnik's constant, Math. Scand. 41 (1977), no. 1, 45–62) and Huxley (Large values of Dirichlet polynomials. III. Acta Arith. 26 (1974/75), no. 4, 435–444). See also the derivation of Theorem 2 in Iwaniec's paper "On zeros of Dirichlet's L series", 
Invent. Math. 23 (1974), 97–104. The constant 2 here corresponds directly to a potential exceptional character, whose contribution is bounded trivially. As Lucia also mentions, we cannot answer this question if $x$ is very close to $q$.


