Geodesic
On the average number of power residues modulo a~composite number
Izvestiya. Mathematics , Tome 74 (2010) no. 6, pp. 1225-1254

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We study the behaviour of the quantities $a_{n}(q)$ and $b_{n}(q)$, that is, the number of $n$th power residues in the reduced and complete residue systems modulo a composite number $q$, respectively, where $n\geqslant2$ is an arbitrary fixed number. In particular, we prove asymptotic formulae for the sum functions $A_{n}(x)$ and $B_{n}(x)$ of these quantities.
Keywords: power residues, average number of power residues, Lehmer–Landau problem. 
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     author = {M. A. Korolev},
     title = {On the average number of power residues modulo a~composite number},
     journal = {Izvestiya. Mathematics },
     pages = {1225--1254},
     publisher = {mathdoc},
     volume = {74},
     number = {6},
     year = {2010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2010_74_6_a4/}
}
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M. A. Korolev. On the average number of power residues modulo a~composite number. Izvestiya. Mathematics , Tome 74 (2010) no. 6, pp. 1225-1254. http://geodesic.mathdoc.fr/item/IM2_2010_74_6_a4/