On the uniform distribution and uniform summability of positive valued multiplicative functions

Abstract. Let \(n\mapsto g(n)\) be a positive valued arithmetic function which tends to infinity as \(n\to\infty\). Following [1], we shall say that the values of \(g\) are uniformly distributed in \((0,\infty)\) if there exists a positive \(c\) such that $$ N(x, g):=\#\bigl\{n: g(n)\leq x\bigr\}\sim cx $$ as \(x\rightarrow\infty\).
In [4] we introduced the class \(\mathcal L^{*}\) of uniformly summable functions \(f\in\mathcal L^{*}\) in case $$ \lim\limits_{K\rightarrow\infty} \sup\limits_{N\geq1}\frac{1}{N}\sum\limits_{n\leq N}|f(n)|<\infty. $$
Here we investigate the asymptotic behaviour of \(N(x,g)\) as \(x\rightarrow\infty\) for multiplicative functions \(g\) such that the associated function \(n\mapsto n/g(n)\) is uniformly summable, and compare it with the behaviour of \(\sum_{n\leq x}n/g(n)\) as \(x\rightarrow\infty\).