Graham, S. W.; Vaaler, Jeffrey D. 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.] Reviewer: George Greaves (Cardiff) Cited in 5 Documents MSC: 11N35 Sieves 42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type Keywords:arithmetic version; large sieve inequality; unweighted form; Fourier transforms Citations:Zbl 0483.42007; Zbl 0541.00002 PDFBibTeX XML