On sums of arithmetic functions involving the greatest common divisor

Abstract

Let gcd(d 1 , . . . , d k ) be the greatest common divisor of the positive integers d 1 , . . . , d k , for any integer k ≥ 2, and let τ and µ denote the divisor function and the Möbius function, respectively. For an arbitrary arithmetic function g and for any real number x > 5 and any integer k ≥ 3, we define the sumIn this paper, we give asymptotic formulas for S τ,k (x) and S µ,k (x) for k ≥ 3.a k b=n µ * g(a)τ k (b).

Hyperbolic summation for functions of the GCD and LCM of several integers

“…in the case of certain special functions f and for k ≥ 2, in particular for k = 2, were given by Heyman [4], Heyman and Tóth [5], Kiuchi and Saad Eddin [10], Krätzel, Nowak and Tóth [12]. Some related probabilistic properties were studied by Iksanov, Marynych and Raschel [8].…”

“…For f (n) = τ 2 (n) =: τ (n) and k = 2 see [4], [5]. For k ≥ 3 the cases of the divisor function τ (n) and the Möbius function µ(n) were studied in [10], giving explicit error terms, and computing the main terms for k = 3 and k = 4. However, note that for f (n) = τ (n), by (1.1),…”