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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 k and
be a weighted count of primes in the arithmetic progression a mod k. We have
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
Then we have
for 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 for some B depending on A, and the result of Davenport–Halberstam has B = A + 5.
See also
References
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