# nt.number theory – Verification of Improved constants in Brun-Titchmarsh theorem

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.

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.