Barban–Davenport–Halberstam theorem

Barban–Davenport–Halberstam theorem In mathematics, the Barban–Davenport–Halberstam theorem is 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.

Statement Let a be coprime to q and {displaystyle vartheta (x;q,a)=sum _{pleq x,;,pequiv a{bmod {q}}}log p } be a weighted count of primes in the arithmetic progression a mod q. We have {displaystyle vartheta (x;q,a)={frac {x}{varphi (q)}}+E(x;q,a) } where φ is Euler's totient function and the error term E is small compared to x. We take a sum of squares of error terms {displaystyle V(x,Q)=sum _{qleq Q}sum _{a{bmod {q}}}|E(x;q,a)|^{2} .} Then we have {displaystyle V(x,Q)=O(Qxlog x)+O(x^{2}(log x)^{-A}) } for {displaystyle 1leq Qleq x} and every positive A, where O is Landau's Big O notation.

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

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