The TME-EMT project
Theorem 2 of [Montgomery & Vaughan, 1973 †Montgomery, H.L., & Vaughan, R.C. 1973
The large sieve
Mathematika, 20(2), 119--133.
] contains the
following explicit version of the Brun-Tichmarsh Theorem.
Let x and y be positive real numbers, and let k and ℓ be relatively
prime positive integers. Then
π(x+y;k,ℓ)−π(x;k,ℓ)<2yϕ(k)log(y/k) provided only that y>k.
Here as usual, we have used the notation
i.e. the number of primes up to z
that are coprime to ℓ
Here is a bound concerning a sieve of dimension 2 proved by
[Siebert, 1976 †Siebert, H. 1976
Montgomery's weighted sieve for dimension two
Monatsh. Math., 82(4), 327--336.
be coprime integers with 2|ab
. Then we have, for x>1
Last updated on July 18th, 2013, by Olivier Ramaré