Asymptotic distribution of normalized arithmetical functions

Abstract:

Let $f(n)$ be an arbitrary arithmetical function and let ${A_N}$ and ${B_N}$ be sequences of real numbers with $0 < {B_N} \to + \infty$ with $N$. We give a sufficient condition for $(f(n) - {A_N})/{B_N}$ to have a limiting distribution. The case when $f(n)$ is defined by $f(n) = \Sigma g(d)$, where the summation is over all divisors $d$ of $n$ and $g(d)$ is any given arithmetical function, is discussed in more detail. A concrete example is given as an application of our result, in which example $f(n)$ is neither additive nor multiplicative. Our method of proof is to approximate $f(n)$ by a suitably chosen additive function, as proposed in [4], and then to apply general theorems available for additive functions.