On averages of completely multiplicative functions over co-prime integer pairs

TL;DR

This paper extends recent results on the averages of completely multiplicative functions from Gaussian integers to co-prime integer pairs, exploring asymptotic behaviors, ergodic averages, and multilinear convergence over primitive lattice points.

Contribution

It provides the analogue of key theorems for Gaussian integers in the setting of co-prime integer pairs and investigates convergence of multilinear averages over primitive lattice points.

Findings

Derived asymptotic formulas for averages over co-prime pairs.

Established ergodic average results along prime factors.

Proved convergence of multilinear averages over primitive lattice points.

Abstract

Recently, Donoso, Le, Moreira and Sun studied the asymptotic behavior of the averages of completely multiplicative functions over the Gaussian integers. They derived Wirsing's theorem for Gaussian integers, answered a question of Frantzikinakis and Host for sum of two squares, and obtained a variant of a theorem of Bergelson and Richter on ergodic averages along the number of prime factors of integers. In this paper, we will show the analogue of these results for co-prime integer pairs. Moreover, building on Frantzikinakis and Host's results, we obtain some convergences on the multilinear averages of multiplicative functions over primitive lattice points.