On certain sums of arithmetic functions involving the gcd and lcm of two positive integers

TL;DR

This paper derives asymptotic formulas with error estimates for sums involving gcd and lcm of two positive integers, focusing on specific arithmetic functions like divisor count, logarithm, and prime factor counts.

Contribution

It provides new asymptotic formulas for hyperbolic sums involving gcd and lcm with various arithmetic functions, including a generalized function unifying several prime factor functions.

Findings

Asymptotic formulas with remainder terms for sums involving gcd and lcm.

Results for specific functions: τ(n), log n, ω(n), Ω(n).

Introduction of a generalized arithmetic function with corresponding asymptotic behavior.

Abstract

We obtain asymptotic formulas with remainder terms for the hyperbolic summations $\sum_{mn\le x} f((m,n))$ and $\sum_{mn\le x} f([m,n])$, where $f$ belongs to certain classes of arithmetic functions, $(m,n)$ and $[m,n]$ denoting the gcd and lcm of the integers $m,n$. In particular, we investigate the functions $f(n)=\tau(n), \log n, \omega(n)$ and $\Omega(n)$. We also define a common generalization of the latter three functions, and prove a corresponding result.