On Vaughan's approximation: The first moment

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

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