Functions of slow increase and integer sequences

Jakimczuk, Rafael

Journal of integer sequences, Tome 13 (2010) no. 1

Résumé

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.

Détail

Classification : 11B99, 11N45
Keywords: functions of slow increase, integer sequences, asymptotic formulas