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Extremal functions for the Fourier transform and the large sieve. (English) Zbl 0552.10028

Topics in classical number theory, Colloq. Budapest 1981, Vol. I, Colloq. Math. Soc. János Bolyai 34, 599-615 (1984).
For each prime \(p\leq Q\) remove \(\omega(p)\) residue classes from the integers in \([M+1,M+N]\), and suppose that \(Z\) integers remain. The authors give a discussion of the arithmetic version of the large sieve inequality in both the unweighted form \[ Z\leq (N-1+Q^ 2)/\{\sum_{1\leq d\leq Q}g(d)\}\] and in the weighted form \[Z\leq 1/\{\sum_{1\leq d\leq Q}g(d)/(N- 1+cdQ)\}.\] Here \(c\) is a positive constant and \(g\) is the multiplicative function given by \(g(p)=\omega (p)/\{p-\omega (p)\};\) \(g(p^{\alpha})=0\) if \(\alpha\geq 2\).
Their discussion does not involve the usual large sieve inequality for trigonometric polynomials but proceeds using Fourier transforms. For the unweighted result they use a function in \(L^1(R)\), featuring in Selberg’s treatment of the large sieve inequality, whose Fourier transform \(\hat F(t)\) is supported on \(| t| \leq \delta\). For the weighted version they use an analogous function in \(L^2(R)\) whose existence is to be established in a subsequent publication.
[For the entire collection see Zbl 0541.00002.]

MSC:

11N35 Sieves
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type