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 states that $$pi(x,q,a)leq frac{(2+o(1))x}{phi(q)log(x/q)}$$ for all $q<x$.

Set $theta=frac{log q}{log x}$. The Brun-Titchmarsh theorem may be rewritten as $$pi(x,q,a)leq(C(theta)+o(1)) frac{x}{phi(q)log x},$$ where $C(theta)=frac{2}{1-theta}$. Depending on the range of $theta$, there have been improved bounds on $C(theta).$ Fouvry, Theorem 3 proved the following bounds:

However, the paper is written in a language unfamiliar to me. Baker-Harman corrected Fouvry’s bounds till $theta=5/7$ in their paper. I want to know if larger bounds on $C(theta)$ were correctly proved in Fouvry. Thanks in advance.