On Vaughan's approximation: The first moment

We investigate the first moment of the difference between $\psi (x;q,a)$ and Vaughan’s approximation, in a certain range of $q$. We show that this last approximation is significantly more precise than the classical $x\mathrm{/}\varphi (q)$, and that it captures the discrepancies of the distribution of primes in arithmetic progressions found in an earlier paper of the author.