ON A SUM INVOLVING CERTAIN ARITHMETIC FUNCTIONS ON PIATETSKI-SHAPIRO AND BEATTY SEQUENCES

Let c, alpha, beta is an element of R be such that 1 < c < 2, alpha > 1 is irrational and with bounded partial quotients, beta is an element of[0,alpha). In this paper, we study asymptotic behaviour of the summations of the form Sigma(n <= N) f([n(c) ])/[n(c) ] and Sigma(n <= N) f([alpha n + beta])/[alpha n+beta], where f is the Euler totient function phi, Dedekind function psi, sum-of-divisors function sigma, or the alternating sum-of-divisors function sigma(alt).