MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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)?

share|cite|improve this question
I don't know of any such cut and dried reference but I find dealing with von-Mangoldt easier than dealing with primes. – Idoneal Oct 26 '10 at 4:29
Yes, that's why the von Mangoldt function was ever defined. At the same time, we sometimes have to deal with primes! – H A Helfgott Oct 26 '10 at 8:59
Well said Mukherjee! – Idoneal Oct 26 '10 at 10:36
This is obviously a cultural reference I am missing. At any rate, can we get back to the question? – H A Helfgott Oct 26 '10 at 11:58
Have you seen Theorem 17.5 of Iwaniec-Kowalski? I think that is all you need. – Idoneal Oct 26 '10 at 13:33

Theorem 17.5 of Iwaniec-Kowalski seems to do the job.

share|cite|improve this answer
For the benefit of anyone else reading - what exactly is Iwaniec-Kowalski? – Gerry Myerson Oct 26 '10 at 23:07
It is the modern bible of analytic number theory.… – Idoneal Oct 27 '10 at 4:25
Thanks. To save others the trouble of clicking through, it's Henryk Iwaniec and Emmanuel Kowalski, Analytic Number Theory, American Mathematical Society Colloquium Publications Volume 53. – Gerry Myerson Oct 27 '10 at 5:22

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.