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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)

Here as usual, we have used the notationLetx andy be positive real numbers, and letk andℓ be relatively prime positive integers. Thenπ(x+y;k,ℓ)−π(x;k,ℓ)<2yϕ(k)log(y/k) provided only thaty>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)

Leta andb be coprime integers with2|ab . Then we have, forx>1 ,∑p≤x,ap+b prime1≤16ωx(logx)2∏p|ab,p>2p−1p−2ω=∏p>2(1−(p−1)−2).

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