Tools on Fourier transforms

1. The large sieve inequality
The best version of the large sieve inequality from [Montgomery & Vaughan, 1974 †Montgomery, H.L., & Vaughan, R.C. 1974
Hilbert's inequality
J. Lond. Math. Soc., II Ser., 8, 73--82.
] and [Montgomery & Vaughan, 1973 †Montgomery, H.L., & Vaughan, R.C. 1973
The large sieve
Mathematika, 20(2), 119--133.
] (obtained at the same time by A. Selberg) is as follows.
Theorem (1974)
Let M$M$ and N1$N\ge 1$ be two real numbers. Let X$X$ be a set of points of [0,1)$[0,1)$ such that
minx,yXminkZ|xy+k|δ>0.
Then, for any sequence of complex numbers (an)M<nM+N$(a_n)_{M < n\le M+N}$, we have
xXM<nM+Nanexp(2iπnx)2M<nM+N|an|2(N1+δ1).
It is very often used with part of the Farey dissection.
Theorem (1974)
Let M$M$ and N1$N\ge 1$ be two real numbers. Let Q1$Q\ge1$ be a real parameter. For any sequence of complex numbers (an)M<nM+N$(a_n)_{M < n\le M+N}$, we have
qQamodq,(a,q)=1M<nM+Nanexp(2iπna/q)2M<nM+N|an|2(N1+Q2).
The summation over a$a$ runs over all invertible classes a$a$ modulo q$q$.

Last updated on July 14th, 2013, by Olivier Ramaré