The Barban-Davenport-Halberstam theorem gives a bound for the average (in L_2 norm) difference between $\sum_{n\leq N: n\equiv a \mod q} \Lambda(n)$ and $N/\phi(q)$. It is obvious that a similar result should hold for the difference between $\sum_{p\leq N: p\equiv a \mod q} 1$ (where $p$ ranges only across primes) and $\pi(N)/\phi(q)$. Does anybody know where in the literature a statement in that form can be found (so that it can be quoted without any further ado - the alternative is to spend some space in its derivation)?
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Theorem 17.5 of Iwaniec-Kowalski seems to do the job. |
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