On the lcm-sum function over arbitrary sets of integers

Reddy, P. Anantha ; Ganeshwar Rao, M. ; Rama Prasad, V. Siva

Serdica Mathematical Journal, Tome 47 (2022) no. 3, pp. 179-190

Résumé

Let \(\mathbb{N}\) denote the set of all positive integers. For \(j,n\in \mathbb{N}\), let \((j,n)\) and \([j,n]\) respectively denote their gcd and lcm. If \(S\subseteq \mathbb{N}\) and \(\alpha\) is a real number then define \(L_{S,\alpha}(n)\) to be the sum of \([j,n]^\alpha\), where \(j\in \{1,2,3,\ldots,n\}\) for which \((j,n)\in S\). In this paper we obtain asymptotic formulae for the summatory functions of \(L_{S,a}(n)\) and \(L_{S,-a}(n)\), where \(a\in \mathbb{N}\) and \(a\geq2\). Apart from deducing some results proved earlier for $S=\mathbb{N}$ by Ikeda and Matsuoka, certain new asymptotic formulae are obtained here.

Détail

Keywords: Zeta-function of \(S\), unitary divisor, \(r\)-free integer, semi-\(r\)-free integer, unitary \(r\)-free integer, \((k,r)\)-integer, 11A25, 11N37