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$x$ and y$y$ be positive real numbers, and let k$k$ and $\ell$ be relatively prime positive integers. Then π(x+y;k,)π(x;k,)<2yϕ(k)log(y/k)$\pi(x+y;k,\ell)-\pi(x;k,\ell) < \frac{2y}{\phi(k)\log (y/k)}$ provided only that y>k$y>k$.
Here as usual, we have used the notation
π(z;k,)=pz,p[k]1,
i.e. the number of primes up to z$z$ that are coprime to $\ell$ modulo k$k$.
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$a$ and b$b$ be coprime integers with 2|ab$2|ab$. Then we have, for x>1$x>1$,
px,ap+b prime116ωx(logx)2p|ab,p>2p1p2ω=p>2(1(p1)2).

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