Journal of Integer Sequences, Vol. 13 (2010), Article 10.1.1

Functions of Slow Increase and Integer Sequences

Rafael Jakimczuk
División Matemática
Universidad Nacional de Luján
Buenos Aires


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 $A_n$ of positive integers that satisfy the asymptotic formula $A_n
\sim n^s f(n)$, where $f(x)$ is a function of slow increase.

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

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