Barban–Davenport–Halberstam theorem

2020 Mathematics Subject Classification: Primary: 11N05 [MSN][ZBL]

A statement about the distribution of prime numbers in an arithmetic progression. It is known that in the long run primes are distributed equally across possible progressions with the same difference. Theorems of the Barban–Davenport–Halberstam type give estimates for the error term, determining how close to uniform the distributions are.

Let $a$ be coprime to $k$ and $$ \vartheta(x;a,k) = \sum_{p<x \,;\, p \equiv a \bmod k} \log p $$ be a weighted count of primes in the arithmetic progression $a$ modulo $k$. We have $$ \vartheta(x;a,k) = \frac{x}{\phi(k)} + E(x;a,k) $$ where the error term $E$ is small compared to $x$. We take a square sum of error terms $$ G(x,Q) = \sum_{k < Q} \sum_{a \bmod k} E^2(x;a,k) . $$ Then we have $$ G(x,Q) = O(Q x \log x) + O(x \log^{-A} x) $$ for any positive $A$.

This form of the theorem is due to Gallagher. The result of Barban is valid only for $Q < x \log^{-B} x$ for some $B$ depending on $A$, and the result of Davenport–Halberstam has $B = A+5$.

Related topics: Bombieri–Vinogradov theorem, Elliott–Halberstam conjecture.

References

• Hooley, C. "On theorems of Barban-Davenport-Halberstam type". In Bennett, M. A.; Berndt, B.C.; Boston, N.; Diamond, H.G.; Hildebrand, A.J.; Philipp, W. (edd) Surveys in number theory: Papers from the millennial conference on number theory. (Natick, MA: A K Peters, 2002) pp. 75–108. ISBN 1-56881-162-4. Zbl 1039.11057.