## On the existence of certain distribution functions

##### Abstract

Let $sigma(n)=sum_{d|n} d$ denote the sum-of-divisors function. Davenport [2] showed $n/sigma(n)$ has a continuous distribution function. That is, $D(u):=lim_{xtoinfty}frac{1}{x} sum_{substack{nleq x n/sigma(n) leq u}} 1$ exists for all $uin[0,1]$ and is a continuous function of $u$. Jennings, Pollack, and Thompson [5] established an analogue of Davenport's theorem. They defined the analogous distribution function as
$tilde{D}_f(u):=lim_{xtoinfty}frac{1}{S(f;x)} sum_{substack{nleq x n/sigma(n) leq u}} f(n)$, where $S(f;x):=sum_{nleq x} f(n)$. They showed that for a certain class of real-valued multiplicative functions $f$, $tilde{D}_f(u)$ exists for all $uin[ 0,1]$ and is both continuous and strictly increasing. In this paper, we further generalize the result by replacing the function $n/sigma(n)$ with certain other multiplicative functions $g(n)$. Hence we define $tilde{D}_{f,g}(u):=lim_{xtoinfty}frac{1}{S(f;x)} sum_{substack{nleq x g(n) leq u}} f(n)$.
We show $tilde{D}_{f,g}(u)$ exists for all $u$ in $[0,1]$ and is a continuous function of $u$. Furthermore, if $S:={ nin mathbb{N} : f(n) >0}$, then $tilde{D}_{f,g}(u)$ is strictly increasing on the interior of the closure of $g(S)$.

##### URI

http://purl.galileo.usg.edu/uga_etd/jennings_emily_n_201405_mahttp://hdl.handle.net/10724/30487