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- The large sieve inequality

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)

It is very often used with part of the Farey dissection.LetM andN≥1 be two real numbers. LetX be a set of points of[0,1) such thatThen, for any sequence of complex numbersminx,y∈Xmink∈Z|x−y+k|≥δ>0. (an)M<n≤M+N , we have∑x∈X∣∣∣∣∑M<n≤M+Nanexp(2iπnx)∣∣∣∣2≤∑M<n≤M+N|an|2(N−1+δ−1).

Theorem (1974)

The summation overLetM andN≥1 be two real numbers. LetQ≥1 be a real parameter. For any sequence of complex numbers(an)M<n≤M+N , we have∑q∈Q∑amodq,(a,q)=1∣∣∣∣∑M<n≤M+Nanexp(2iπna/q)∣∣∣∣2≤∑M<n≤M+N|an|2(N−1+Q2).

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