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(Received: 15 August 2020. Received in revised form: 31 August 2020. Accepted: 3 September 2020. Published online: 5 October 2020.) c © 2020 the authors. This is an open access article under the CC BY (International 4.0) license (www.creativecommons.org/licenses/by/4.0/). Abstract Let gcd(k, j) be the greatest common divisor of the positive integers k and j. For any real number x > 1 and for any fixed positive integers s and r, we give an asymptotic formula of the sum function ∑

An asymptotic formula of a sum function involving gcd and characteristic function of the set of r–free numbers