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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)].
##### MSC:
 11N56 Rate of growth of arithmetic functions 11L07 Estimates on exponential sums