On the Sums Running over Reduced Residue Classes Evaluated at Polynomial Arguments

Abstract

For a given polynomial G we study the sums φ_m(n) := ∑′k^m and φ_G(n) = ∑′G(k) where m ≥ 0 is a fixed integer and ∑′ runs through all integers k with 1 ≤ k ≤ n and gcd(k, n) = 1. Although, for m ≥ 1 the function φ_m is not multiplicative, analogue to the Euler function, we obtain expressions for φ_m(n) and φ_G(n). Also, we estimate the averages ∑_n≤x φ_m(n) and ∑_n≤xφ_G(n), the alternative averages ∑_n≤x(−1)ⁿ⁻¹φ_m(n) and ∑_n≤x(−1)ⁿ⁻¹φ_G(n).