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

For a given polynomial G we study the sums phi(m)(n) := Sigma'k(m) and phi(G)(n) = Sigma'G(k) where m >= 0 is a fixed integer and Sigma' runs through all integers k with 1 <= k <= n and gcd(k, n) = 1. Although, for m >= 1 the function phi(m) is not multiplicative, analogue to the Euler function, we obtain expressions for phi(m)(n) and phi(G)(n). Also, we estimate the averages Sigma(n <= x)phi(m)(n) and Sigma(n <= x)phi(G)(n), the alternative averages Sigma(n <= x)(-1)(n-1)phi(m)(n) and Sigma(n <= x)(-1)(n-1)phi(G)(n).