an:02010135
Zbl 1028.11059
Arkhipov, G. I.; Chubarikov, V. N.
On the greatest prime factor of a sequence of the form \([n^c]\)
RU
Math. Montisnigri 8, 17-31 (1997).
00101759
1997
j
11N56 11L07
greatest prime factor; Vinogradov's method of trigonometric sums
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 \textit{C. Hooley}, Acta Math. 117, 281--299 (1967; Zbl 0146.05704)].
Ivan D. Chipchakov (Sofia)
Zbl 0146.05704