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On the greatest prime factor of a sequence of the form \([n^c]\). (Russian) Zbl 1028.11059
Let \(c\) be a fixed real number satisfying \(4/3<c<2\), and for each real number \(x\geq 2\), let \(y(x)\) be the greatest prime factor of the product \(\prod[n^c]\), taken over the set of positive integers \(n\leq x\). The authors show that \(y(x)>x^{(27-13c)/28}\). The proof relies essentially on Vinogradov’s method of trigonometric sums. It is known that if \(c=2\), then \(y(x)\) satisfies the stronger inequality \(y(x)\geq x^{11/10}\) [see C. Hooley, Acta Math. 117, 281–299 (1967; Zbl 0146.05704)].
11N56 Rate of growth of arithmetic functions
11L07 Estimates on exponential sums
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