On the Sums Running over Reduced Residue Classes Evaluated at Polynomial Arguments (Contemporary Mathematics, 2020)
For a given polynomial G we study the sums φm(n) := ∑′km and φG(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 φm is not multiplicative, analogue to the Euler function, we obtain expressions for φm(n) and φG(n). Also, we estimate the averages ∑n≤x φm(n) and ∑n≤xφG(n), the alternative averages ∑n≤x(−1)n−1φm(n) and ∑n≤x(−1)n−1φG(n).
Keywords: Euler function, residue systems, arithmetic function, alternating sum