Functions of Slow Increase and Integer Sequences

We study some properties of functions that satisfy the condition $f'(x)=o\left(\frac{f(x)}{x}\right)$ , for $x\rightarrow \infty$ , i.e., $\lim_{x\rightarrow \infty}\frac{ f'(x)}{\frac{f(x)}{x}}= 0$ . We call these ``functions of slow increase'', since they satisfy the condition $\lim_{x\rightarrow \infty}\frac{f(x)}{x^{\alpha}} =0$ for all $\alpha>0$ . A typical example of a function of slow increase is the function $f(x)= \log x$ . As an application, we obtain some general results on sequence

of positive integers that satisfy the asymptotic formula $A_n \sim n^s f(n)$ , where

is a function of slow increase.

Received September 14 2009; revised version received December 21 2009. Published in Journal of Integer Sequences, December 23 2009.

Functions of Slow Increase and Integer Sequences

Rafael Jakimczuk División Matemática Universidad Nacional de Luján Buenos Aires Argentina

Rafael Jakimczuk
División Matemática
Universidad Nacional de Luján
Buenos Aires
Argentina