ACTA ARITHMETICA 136.1 (2009) Arithmetic functions on Beatty sequences by Alex G. Abercrombie (Pembroke), William D. Banks (Columbia, MO) and Igor E. Shparlinski (Sydney) 1. Introduction 1.1. Background. For a real number α> 1, the homogeneous Beatty sequence corresponding to α is the sequence of natural numbers given by B α =(⌊αk⌋) k∈N , where ⌊t⌋ denotes the greatest integer ≤ t. Beatty sequences appear in a variety of contexts and have been extensively explored in the literature. In particular, summatory functions of the form (1) S α (f,x)=  n≤x, n∈Bα f (n) have been studied when the arithmetic function f is • a multiplicative or an additive function (see [1, 2, 8, 9, 10, 11]); • a Dirichlet character (see [2, 3, 5]); • the characteristic function of primes or smooth numbers (see [4, 6, 7]). For an arbitrary arithmetic function f we define (2) S (f,x)= S 1 (f,x)=  n≤x f (n). Abercrombie [1] has shown that for the divisor function τ the asymptotic formula (3) S α (τ,x)= α -1 S (τ,x)+ O(x 5/7+ε ) holds for any ε> 0 and almost all α> 1 (with respect to Lebesgue measure), where the implied constant depends only on α and ε. This result has been 2000 Mathematics Subject Classification : 11B83, 11J83, 11K65. Key words and phrases : Beatty sequence, arithmetic function. DOI: 10.4064/aa136-1-6 [81] c  Instytut Matematyczny PAN, 2009