Sieve bounds

1. Some upper bounds
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.
Theorem (1973)
Let x and y be positive real numbers, and let k and be relatively prime positive integers. Then π(x+y;k,)π(x;k,)<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 modulo k.
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.
Theorem (1976)
Let a and b be coprime integers with 2|ab. Then we have, for x>1,
px,ap+b prime116ωx(logx)2p|ab,p>2p1p2ω=p>2(1(p1)2).

Last updated on July 18th, 2013, by Olivier Ramaré