On the number of $k$-compositions of $n$ satisfying certain coprimality conditions

TL;DR

This paper extends and refines asymptotic estimates for counting k-compositions of n with coprimality constraints, improving error terms using multiplicative functions and partition formulas.

Contribution

It introduces a new approach based on multiplicative arithmetic functions and asymptotic formulas to improve estimates for coprime k-compositions.

Findings

Refined asymptotic estimates with better error bounds

New method using multiplicative functions and partition formulas

Generalization of previous results by Bubboloni, Luca, and Spiga

Abstract

We generalize the asymptotic estimates by Bubboloni, Luca and Spiga (2012) on the number of $k$-compositions of $n$ satisfying some coprimality conditions. We substantially refine the error term concerning the number of $k$-compositions of $n$ with pairwise relatively prime summands. We use a different approach, based on properties of multiplicative arithmetic functions of $k$ variables and on an asymptotic formula for the restricted partition function.