On a sum involving small arithmetic functions

Abstract

Let [Formula: see text] be any arithmetic function and define [Formula: see text]. If the function [Formula: see text] is small, namely, [Formula: see text] then the error term in the asymptotic formula of [Formula: see text] has the form [Formula: see text] In this paper, we shall prove that if a small arithmetic function [Formula: see text] has a binary additive property, then we can break the barrier [Formula: see text] in the asymptotic formula of [Formula: see text].