\documentclass[12pt,reqno]{article} \usepackage[usenames]{color} \usepackage{amssymb} \usepackage{graphicx} \usepackage{amscd} \usepackage[colorlinks=true, linkcolor=webgreen, filecolor=webbrown, citecolor=webgreen]{hyperref} \definecolor{webgreen}{rgb}{0,.5,0} \definecolor{webbrown}{rgb}{.6,0,0} \usepackage{color} \usepackage{fullpage} \usepackage{float} \usepackage{psfig} \usepackage{graphics,amsmath,amssymb} \usepackage{amsthm} \usepackage{amsfonts} \usepackage{latexsym} \usepackage{epsf} \setlength{\textwidth}{6.5in} \setlength{\oddsidemargin}{.1in} \setlength{\evensidemargin}{.1in} \setlength{\topmargin}{-.5in} \setlength{\textheight}{8.9in} \newcommand{\seqnum}[1]{\href{http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=#1}{\underline{#1}}} \begin{document} \begin{center} \epsfxsize=4in \leavevmode\epsffile{logo129.eps} \end{center} \begin{center} \vskip 1cm{\LARGE\bf Functions of Slow Increase\\ \vskip .1in and Integer Sequences} \vskip 1cm \large Rafael Jakimczuk\\ Divisi\'on Matem\'atica \\ Universidad Nacional de Luj\'an\\ Buenos Aires\\ Argentina\\ \href{mailto:jakimczu@mail.unlu.edu.ar}{\tt jakimczu@mail.unlu.edu.ar}\\ \end{center} \vskip .2in \begin{abstract} 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. \end{abstract} \newtheorem{theorem}{Theorem} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{defin}[theorem]{Definition} \newenvironment{definition}{\begin{defin}\normalfont\quad}{\end{defin}} \newtheorem{examp}[theorem]{Example} \newenvironment{example}{\begin{examp}\normalfont\quad}{\end{examp}} \newtheorem{rema}[theorem]{Remark} \newenvironment{remark}{\begin{rema}\normalfont\quad}{\end{rema}} \section{Functions of Slow Increase} \begin{definition}\label{D1} Let $f(x)$ be a function defined on the interval $[a, \infty)$ such that $f(x)>0$, $\lim_{x\rightarrow \infty} f(x)=\infty $ and with continuous derivative $f'(x)>0$. The function $f(x)$ is {\em of slow increase} if the following condition holds. \begin{equation}\label{A1} \lim_{x\rightarrow \infty}\frac{ f'(x)}{\frac{f(x)}{x}}= 0. \end{equation} \end{definition} Typical functions of slow increase are $f(x)=\log x$, $f(x)=\log^2 x$, $f(x)=\log\log x$, $f(x)=\frac{\log x}{\log\log x}$ and $\Psi:(0,\infty)\rightarrow (0,\infty)$, $\Psi(x)= \frac{\Gamma'(x)}{\Gamma(x)}$, where $\Gamma(x)=\int^{\infty}_{0}t^{x-1}e^{-t}\ dt$, which generalize the harmonic sum $H_n: N^{*} \rightarrow R$, $H_n=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}$ to $(0,\infty)$, namely $H_n=\Psi(n+1)+\gamma$, where $\gamma$ is Euler's constant. We have the following theorems. \begin{theorem}\label{T2} If $f(x)$ and $g(x)$ are functions of slow increase and $C$ and $\alpha$ are positive constants then the following functions are of slow increase. \[f(x)+C, \qquad f(x)-C, \qquad C f(x),\qquad f(x)g(x),\qquad f(x)^{\alpha},\] \[f(g(x)),\qquad \log f(x),\qquad f(x^{\alpha}),\qquad f(x^{\alpha}g(x)),\qquad f(x)+g(x).\] If $f(x)$ and $g(x)$ are functions of slow increase, $\lim_{x\rightarrow \infty}\frac{f(x)}{g(x)}=\infty$ and $\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right)>0$ then $\frac{f(x)}{g(x)}$ is a function of slow increase. If $h(x)$ is a function such that $h(x)>0$, $\lim_{x\rightarrow \infty} h(x)=\infty $ and with continuous derivative $h'(x)>0$, then $h(\log x)$ is a function of slow increase if and only if $\lim_{x\rightarrow \infty}\frac{h'(x)}{h(x)}=0$. If $h(x)$ is a function such that $h(x)>0$, $\lim_{x\rightarrow \infty} h(x)=\infty $ and with continuous derivative $h'(x)>0$, then $e^{h(x)}$ is a function of slow increase if and only if $\lim_{x\rightarrow \infty}x h'(x)=0$. If $f(x)$ is a function of slow increase the following limit holds. \begin{equation}\label{A2} \lim_{x\rightarrow \infty}\frac{\log f(x)}{\log x}=0. \end{equation} \end{theorem} \begin{proof} Use Definition~\ref{D1}. \end{proof} \begin{theorem}\label{T3} The function $f(x)$ is of slow increase if and only if $\frac{f(x)}{x^{\alpha}}$ has negative derivative (from a certain $x_{\alpha}$) for all $\alpha>0$. \end{theorem} \begin{proof} We have \begin{equation}\label{A3} \frac{d}{dx}\left(\frac{f(x)}{x^{\alpha}}\right)=\frac{f(x)}{x^{\alpha+1}}\left(\frac{xf'(x)}{f(x)}-\alpha\right). \end{equation} Therefore if limit (\ref{A1}) holds we obtain that for all $\alpha>0$, \begin{equation}\label{A4} \frac{d}{dx}\left(\frac{f(x)}{x^{\alpha}}\right)< 0 \end{equation} for $x> x_{\alpha}$. On the other hand, if (\ref{A4}) holds (for $x> x_{\alpha}$), (\ref{A3}) gives \[0<\frac{xf'(x)}{f(x)}<\alpha.\] Consequently we obtain (note that $\alpha$ is arbitrary) \[\lim_{x\rightarrow \infty}\frac{x f'(x)}{f(x)}= 0.\] That is, equation (\ref{A1}). \end{proof} The following theorem justifies the term ``slow increase''. \begin{theorem}\label{T4} If the function $f(x)$ is of slow increase then \begin{equation}\label{A5} \lim_{x\rightarrow \infty}\frac{f(x)}{x^{\beta}} =0 \end{equation} for all $\beta>0$. \end{theorem} \begin{proof} Let $\alpha>0$ be such that $\alpha<\beta$. Then $\frac{f(x)}{x^{\alpha}}$ has a negative derivative (for $x>x_{\alpha}$), then it is decreasing, therefore it is bounded $0< \frac{f(x)}{x^{\alpha}} -1$ and for all $\beta$. \end{theorem} \begin{proof} We have \begin{equation}\label{A9} \sum^{\infty}_{i=1}i^{\alpha}f(i)^{\beta}= \sum^{\infty}_{i=1}\left(i^{\alpha+1}f(i)^{\beta}\right)\frac{1}{i}. \end{equation} Now, it is well-known that \begin{equation}\label{A10} \sum^{\infty}_{i=1}\frac{1}{i}=\infty. \end{equation} On the other hand, we have (note that $\alpha +1>0$) \begin{equation}\label{A11} \lim_{i\rightarrow \infty}i^{\alpha+1}f(i)^{\beta}=\infty. \end{equation} Limit (\ref{A11}) is clearly true if $\beta \geq 0$. If $\beta < 0$ limit (\ref{A11}) is a direct consequence of Theorem~\ref{T2} ($f(x)^{-\beta}$ is of slow increase) and Theorem~\ref{T4}. Finally, equations (\ref{A9}), (\ref{A10}) and (\ref{A11}) give equation (\ref{A8}). \end{proof} \begin{theorem} If the function $f(x)$ is of slow increase then the following limit holds \begin{equation}\label{A12} \lim_{x\rightarrow \infty}\frac{\int^{x}_{a}t^{\alpha}f(t)^{\beta}\ dt}{\frac{x^{\alpha+1}}{\alpha+1}f(x)^{\beta}} =1 \end{equation} for all $\alpha> -1$ and for all $\beta$. \end{theorem} \begin{proof} We have (see (\ref{A11})) \[\lim_{x\rightarrow \infty}\frac{x^{\alpha+1}}{\alpha+1}f(x)^{\beta}=\infty.\] On the other hand, the function $t^{\alpha}f(t)^{\beta}$ is either increasing or decreasing. Use Theorem~\ref{T2} and Theorem~\ref{T3} in the case $\alpha<0$, $\beta>0$ and $\alpha>0$, $\beta<0$. The others cases are trivial. Consequently (\ref{A8}) implies, \[\lim_{x\rightarrow \infty}\int^{x}_{a}t^{\alpha}f(t)^{\beta}\ dt =\infty.\] Now, limit (\ref{A12}) is a direct consequence of the L'Hospital's rule and limit (\ref{A1}). \end{proof} Some particular cases of this theorem are the following: If $\alpha= 0$ we have \begin{equation}\label{A13} \int^{x}_{a}f(t)^{\beta}\ dt \sim x f(x)^{\beta}. \end{equation} If $\alpha= 0$ and $\beta=1$ we have \begin{equation}\label{A14} \int^{x}_{a}f(t)\ dt \sim x f(x). \end{equation} If $\alpha= 0$ and $\beta=-1$ we have \begin{equation}\label{A15} \int^{x}_{a}\frac{1}{f(t)}\ dt \sim \frac{x}{f(x)}. \end{equation} \begin{theorem}\label{T8} If the function $f(x)$ is of slow increase and $C$ is a constant then the following limit holds \begin{equation}\label{A16} \lim_{x\rightarrow \infty}\frac{f(x+C)}{f(x)} =1. \end{equation} \end{theorem} \begin{proof} If $C>0$, applying the Lagrange's Theorem we obtain \begin{equation}\label{A17} 0\leq \frac{f(x+C)-f(x)}{f(x)}= \frac{C f'(\xi)}{f(x)}, \qquad (x<\xi0 $ then the following limit holds \begin{equation}\label{A18} \lim_{x\rightarrow \infty}\frac{f(C x)}{f(x)} =1. \end{equation} \end{theorem} \begin{proof} Suppose that $C>1$. Applying Lagrange's theorem we obtain \begin{equation}\label{A19} 0\leq \frac{f(Cx)-f(x)}{f(x)}= \frac{(Cx-x) f'(\xi)}{f(x)}\leq (C-1)\frac{x f'(x)}{f(x)}, \quad (x<\xi1) \end{equation} and $f(x)$ is a function of slow increase. Let $\psi(x)$ be the number of $A_n$ that do not exceed $x$. \begin{example}\label{E11} If $A_n=p_n$ is the sequence of prime numbers we have (Prime Number Theorem) $s=1$ and $f(x)= \log x$. If $A_n = c_{n,k}$ is the sequence of numbers with $k$ prime factors we have $s=1$ and $ f(x) = \frac{(k-1)!\log x}{(\log\log x)^{k-1}}$ (see \cite {Jakim2}). If $A_n=p_n^2$ we have $s=2$ and $f(x)= \log^2 x$. \end{example} \begin{remark} Note that: (i) Theorem~\ref{T4} implies that $s\geq 1$ in equation (\ref{A23}). (ii) There exists a strictly increasing sequence $A_n$ that satisfies (\ref{A23}), for example $A_n= \lfloor n^s f(n) \rfloor$. (iii) If the function $g(x)$ is of slow increase then $\frac{g(A_n)}{g(n)}\rightarrow l\Leftrightarrow \frac{g(n^s f(n))}{g(n)}\rightarrow l$ and $\frac{g(A_n)}{g(n)}\rightarrow \infty \Leftrightarrow \frac{g(n^s f(n))}{g(n)}\rightarrow \infty$, because (Theorem~\ref{T10}) $g(A_n)\sim g(n^s f(n))$. \end{remark} \begin{theorem} If $A_n$ satisfies (\ref{A23}) and $g(x)$ is a function of slow increase then the following equations hold \begin{equation}\label{A24} A_{n+1}\sim A_n, \end{equation} \begin{equation}\label{A25} \lim_{n\rightarrow \infty}\frac{A_{n+1}-A_n}{A_n} =0, \end{equation} \begin{equation}\label{A26} \log A_{n+1}\sim \log A_n, \end{equation} \begin{equation}\label{A27} g(A_{n+1})\sim g(A_n), \end{equation} \begin{equation}\label{A28} \log A_n \sim s \log n, \end{equation} \begin{equation}\label{A29} \log \log A_n \sim \log \log n, \end{equation} \[\lim_{x\rightarrow \infty}\frac{\psi(x)}{x}=0.\] \end{theorem} \begin{proof} Equation (\ref{A24}) is an immediate consequence of equation (\ref{A23}) and Theorem~\ref{T8}. Equation (\ref{A25}) is an immediate consequence of equation (\ref{A24}). Equations (\ref{A26}) and (\ref{A27}) are an immediate consequence of equation (\ref{A24}) and Theorem~\ref{T10}. Equation (\ref{A28}) is an direct consequence of equations (\ref{A23}) and (\ref{A2}). Equation (\ref{A29}) is an direct consequence of (\ref{A28}). The last limit is an immediate consequence of (\ref{A23}) ($(A_n/n) \rightarrow \infty $) and (\ref{A24}). \end{proof} \begin{theorem}\label{T14} If $A_n$ satisfies (\ref{A23}) and $g(x)$ is a function of slow increase then the following equation holds (note that $l\geq 1$). \begin{equation}\label{A30} g (A_n)\sim l g( n) \Leftrightarrow g( \psi(x))\sim \frac{1}{l}g (x). \end{equation} In particular (see (\ref{A28}) and (\ref{A29})) \begin{equation}\label{A31} \log A_n\sim s \log n \Leftrightarrow \log \psi(x)\sim \frac{1}{s}\log x, \end{equation} \begin{equation}\label{A32} \log \log A_n\sim \log \log n \Leftrightarrow \log \log \psi(x)\sim \log \log x. \end{equation} \end{theorem} \begin{proof} We have \begin{eqnarray} g( \psi(x))\sim \frac{1}{l}g( x) &\Rightarrow& g (\psi(A_n))\sim \frac{1}{l}g (A_n) \Rightarrow g( n) \sim \frac{1}{l}g( A_n) \nonumber\\&\Rightarrow& g( A_n) \sim l g( n). \nonumber \end{eqnarray} On the other hand \begin{eqnarray}\label{A33} g( A_n )\sim l g( n) \Rightarrow \ g( A_n) \sim l g (\psi(A_n))\Rightarrow g( \psi(A_n))\sim \frac{1}{l}g( A_n). \end{eqnarray} If $ A_n\leq x< A_{n+1}$ we have \[\frac{g( \psi(A_n))}{\frac{1}{l}g( A_{n+1})}\leq \frac{g( \psi(x))}{\frac{1}{l}g( x)} \leq \frac{g( \psi(A_n))}{\frac{1}{l}g( A_n)}.\] Now, both sides have limit 1 (see (\ref{A33}) and (\ref{A27})). \end{proof} We shall need the following well-known lemma (see \cite[p.\ 332]{ReyPastor5}). \begin{lemma}\label{L15} Let $\sum^{\infty}_{i=1}a_i$ and $\sum^{\infty}_{i=1}b_i$ be two series of positive terms such that $\lim_{i\rightarrow \infty}\frac{a_i}{b_i}=1$. Then if $\sum^{\infty}_{i=1}b_i$ is divergent, the following limit holds. \[\lim_{n\rightarrow \infty}\frac{\sum^{n}_{i=1}a_i}{\sum^{n}_{i=1}b_i}=1.\] \end{lemma} In the following theorem we shall obtain information on $\psi(x)$ when $s=1$ (see (\ref{A23})) and $f(A_n)\sim f(n)$. \begin{theorem} \label{T16} If $f(A_n)\sim f(n)$ then \begin{equation}\label{A34} A_n\sim n f(n) \Leftrightarrow \psi(x)\sim \frac{x}{f(x)}\Leftrightarrow \psi(x)\sim \int^{x}_{a}\frac{1}{f(t)}\ dt \Leftrightarrow \sum_{A_i\leq x}f(A_i)\sim x. \end{equation} Besides if $g(x)$ is a function of slow increase and $g(A_n)\sim l'g(n)$ then \begin{equation}\label{A35} \psi(x)\sim \frac{\sum_{A_i\leq x}g(A_i)^{\beta}}{g(x)^{\beta}} \end{equation} for all $\beta$. \end{theorem} \begin{proof} We have (note that $\frac{x}{f(x)}\rightarrow \infty$, see (\ref{A6})) \begin{eqnarray} \psi(x)\sim \frac{x}{f(x)}&\Rightarrow& \psi(A_n)\sim \frac{A_n}{f(A_n)}\Rightarrow n \sim \frac{A_n}{f(A_n)}\Rightarrow A_n \sim n f(A_n)\nonumber\\&\Rightarrow& A_n \sim n f(n).\nonumber \end{eqnarray} On the other hand \begin{eqnarray}\label{A36} A_n \sim n f(n)&\Rightarrow& A_n \sim \psi(A_n) f(n)\Rightarrow \psi(A_n)\sim \frac{A_n}{f(n)} \nonumber\\&\Rightarrow& \psi(A_n)\sim \frac{A_n}{f(A_n)}. \end{eqnarray} If $ A_n\leq x< A_{n+1}$ we have (note that $\frac{x}{f(x)}$ is increasing, see Theorem~\ref{T3}) \begin{equation}\label{A37} \frac{\psi(A_n)}{\frac{A_{n+1}}{f(A_{n+1})}}\leq \frac{\psi(x)}{\frac{x}{f(x)}}\leq \frac{\psi(A_n)}{\frac{A_n}{f(A_n)}}. \end{equation} Now, both sides have limit 1 (see (\ref{A36}), (\ref{A24}) and (\ref{A27})). Consequently \[A_n\sim n f(n) \Rightarrow \psi(x)\sim \frac{x}{f(x)}.\] On the other hand (see (\ref{A15})) \[\psi(x)\sim \frac{x}{f(x)}\Leftrightarrow \psi(x)\sim \int^{x}_{a}\frac{1}{f(t)}\ dt.\] Note that (see (\ref{A13})) \[\int^{n}_{a}g(x)^{\beta}\ dx \sim n g(n)^{\beta}.\] Therefore as $g(x)^{\beta}$ is either increasing or decreasing, \begin{equation}\label{A38} \sum^{n}_{i=1}g(i)^{\beta}=\int^{n}_{a}g(x)^{\beta}\ dx +h(n)\sim n g(n)^{\beta}. \end{equation} Equation (\ref{A38}), $g(A_n)^{\beta}\sim l'^{\beta} g(n)^{\beta}$ and Lemma~\ref{L15} give \[\sum^{n}_{i=1}g(A_i)^{\beta}\sim n l'^{\beta} g(n)^{\beta}.\] That is \[\sum_{A_i\leq A_n}g(A_i)^{\beta}\sim \psi(A_n) g(A_n)^{\beta}.\] Consequently \begin{equation} \label{A39} \psi(A_n)\sim \frac{\sum_{A_i\leq A_n}g(A_i)^{\beta}}{g(A_n)^{\beta}}. \end{equation} If $ A_n\leq x < A_{n+1}$ we have $(\beta>0)$ \[\frac{\psi(A_n)}{\frac{\sum_{A_i\leq A_n}g(A_i)^{\beta}}{g(A_n)^{\beta}}}\leq \frac{\psi(x)}{\frac{\sum_{A_i\leq x}g(A_i)^{\beta}}{g(x)^{\beta}}}\leq \frac{\psi(A_n)}{\frac{\sum_{A_i\leq A_n}g(A_i)^{\beta}}{g(A_{n+1})^{\beta}}}.\] Now, both sides have limit 1 (see (\ref{A39}) and (\ref{A27})). Therefore \[\psi(x)\sim \frac{\sum_{A_i\leq x}g(A_i)^{\beta}}{g(x)^{\beta}}.\] That is, equation (\ref{A35}). If $\beta<0$, the proof of (\ref{A35}) is the same. Consequently if $g(x)=f(x)$ and $\beta = 1$ we find that \[\psi(x)\sim \frac{x}{f(x)}\Leftrightarrow \sum_{A_i\leq x}f(A_i)\sim x.\] \end{proof} \begin{example} Let us consider the sequence $p_n$ of prime numbers, in this case we have (Prime Number Theorem) $p_n\sim n \log n$ and $\psi(x)=\pi(x)\sim x/(\log x)$. Let us consider the sequence $c_{n,k}$ of numbers with $k$ prime factors, in this case we have $ c_{n,k} \sim \frac{(k-1)!n \log n}{(\log\log n)^{k-1}}$ (see Example~\ref{E11}) and (Landau's Theorem) (see \cite {Hardy1, Jakim2}) $\psi(x)\sim \frac{x (\log\log x)^{k-1}}{(k-1)! \log x}$. \end{example} In the following general theorem we obtain information on $\psi(x)$ if $f(A_n)\sim l f(n)$. Theorem~\ref{T16} is a particular case of this Theorem. \begin{theorem}\label{T18} If $f(A_n)\sim l f(n)$ then \begin{eqnarray} A_n\sim n^s f(n) &\Leftrightarrow& \psi(x)\sim l^{\frac{1}{s}}\frac{x^{\frac{1}{s}}}{f(x)^{\frac{1}{s}}}\Leftrightarrow \psi(x)\sim \frac{l^{\frac{1}{s}}}{s}\int^{x}_{a}\frac{t^{-1+\frac{1}{s}}}{f(t)^{\frac{1}{s}}}\ dt\nonumber\\&\Leftrightarrow& \sum_{A_i\leq x}f(A_i)^{\frac{1}{s}}\sim l^{\frac{1}{s}} x^{\frac{1}{s}}.\nonumber \end{eqnarray} Besides if $g(x)$ is a function of slow increase and $g(A_n)\sim l'g(n)$ then \begin{equation}\label{A40} \psi(x)\sim \frac{\sum_{A_i\leq x}g(A_i)^{\beta}}{g(x)^{\beta}} \end{equation} for all $\beta$. \end{theorem} \begin{proof} The proof that \[A_n\sim n^s f(n) \Leftrightarrow \psi(x)\sim l^{\frac{1}{s}}\frac{x^{\frac{1}{s}}}{f(x)^{\frac{1}{s}}}\] is the same as in Theorem~\ref{T16}. Now, see equation (\ref{A12}), \[\int^{x}_{a}\frac{t^{-1+\frac{1}{s}}}{s f(t)^{\frac{1}{s}}}\ dt \sim \frac{x^{\frac{1}{s}}}{f(x)^{\frac{1}{s}}}.\] Therefore \[\psi(x)\sim l^{\frac{1}{s}}\frac{x^{\frac{1}{s}}}{f(x)^{\frac{1}{s}}}\Leftrightarrow\psi(x)\sim \frac{l^{\frac{1}{s}}}{s}\int^{x}_{a}\frac{t^{-1+\frac{1}{s}}}{f(t)^{\frac{1}{s}}}\ dt. \] The proof of the equation (\ref{A40}) is the same as in Theorem~\ref{T16}. If $g(x)=f(x)$ and $\beta = 1/s$ then we find that \[\psi(x)\sim l^{\frac{1}{s}}\frac{x^{\frac{1}{s}}}{f(x)^{\frac{1}{s}}} \Leftrightarrow \sum_{A_i\leq x}f(A_i)^{\frac{1}{s}}\sim l^{\frac{1}{s}} x^{\frac{1}{s}}.\] \end{proof} \begin{example} Let us consider the following sequence of positive integers (see Theorem~\ref{T22}) \[A_n=\sum^{n}_{i=1}p_i^k \sim \frac{n^{k+1}}{k+1}\log^k n\] where $k$ is a positive integer. In this case we have $s=k+1$, $f(x)= \frac{\log^k x}{k+1}$ and $l= (k+1)^k$. Consequently \[\psi(x)\sim (k+1)\frac{x^{\frac{1}{k+1}}}{\left(\log x\right)^{\frac{k}{k+1}}}.\] \end{example} Let us consider the sequence $P_n$ of the $A_n$ powers. For example, if $A_n=p_n$ is the sequence of prime numbers, $P_n$ is the sequence of prime powers . Let $\lambda(x)$ be the number of $P_n$ that do not exceed $x$. \begin{theorem} If $A_n$ satisfies (\ref{A23}) then \begin{equation}\label{A41} \lambda(x)\sim \psi(x). \end{equation} \end{theorem} \begin{proof} The $A_i\leq x$ are $A_1, A_2,\ldots, A_{\psi(x)}$. Let us write \[A_i^{\alpha_i}= x, \qquad (i=1,2,\ldots, \psi(x)).\] Therefore \[\alpha_i = \frac{\log x}{\log A_i},\qquad(i=1,2,\ldots, \psi(x)).\] We have the following inequalities \begin{equation}\label{A42} \psi(x)\leq \lambda(x)\leq \sum^{\psi(x)}_{i=1}[\alpha_i]\leq \sum^{\psi(x)}_{i=1}\alpha_i = \log x \sum^{\psi(x)}_{i=1}\frac{1}{\log A_i}. \end{equation} Equation (\ref{A28}) gives \begin{equation}\label{A43} \frac{1}{\log A_n}\sim \frac{1}{s \log n}. \end{equation} Note that (see (\ref{A15})) \[\int^{x}_{2}\frac{1}{\log t}\ dt \sim \frac{x}{\log x}.\] Now, \begin{eqnarray}\label{A44} &&\frac{1}{\log A_1}+\sum^{\psi(x)}_{i=2}\frac{1}{s\log i}=\frac{1}{\log A_1}+\frac{1}{s}\sum^{\psi(x)}_{i=2}\frac{1}{\log i}\nonumber\\&=& \frac{1}{s}\int^{\psi(x)}_{2}\frac{1}{\log t}\ dt +O(1) \sim \frac{\psi(x)}{s \log \psi(x)}. \end{eqnarray} Equations (\ref{A43}), (\ref{A44}) and Lemma~\ref{L15} give \begin{equation}\label{A45} \sum^{\psi(x)}_{i=1}\frac{1}{\log A_i}\sim \frac{\psi(x)}{s \log \psi(x)}. \end{equation} Equations (\ref{A42}) and (\ref{A45}) give \[\psi(x)\leq \lambda(x)\leq h(x) \frac{\psi(x)\log x}{s \log \psi(x)},\] where $h(x)\rightarrow 1$. That is \begin{equation}\label{A46} 1\leq \frac{\lambda(x)}{\psi(x)}\leq h(x) \frac{\log x}{s \log \psi(x)}. \end{equation} Finally, equations (\ref{A31}) and (\ref{A46}) give (\ref{A41}). \end{proof} \begin{corollary} The following limit holds. \[\lim_{x\rightarrow \infty}\frac{\sum^{\psi(x)}_{i=1}\left(\alpha_i-[\alpha_i]\right)}{\psi(x)}=0.\] That is, the mean fractional part has limit zero. \end{corollary} \begin{theorem}\label{T22} If $A_n$ satisfies (\ref{A23}) then the following asymptotic formulas hold \begin{equation}\label{A47} \sum^{n}_{i=1}A_i^{\alpha} \sim \frac{n^{s\alpha +1}f(n)^{\alpha}}{s\alpha +1}\sim \frac{n}{s \alpha +1}\ A_n^{\alpha},\qquad (\alpha >0), \end{equation} \begin{equation}\label{A48} \sum_{A_i\leq x}A_i^{\alpha} \sim \frac{\psi (x)}{s \alpha +1}\ x^{\alpha},\qquad (\alpha >0). \end{equation} \end{theorem} \begin{proof} Let us consider the sum \begin{equation}\label{A49} 1+2+\cdots+(n'-1)+\sum^{n}_{i=n'}\left(i^s f(i)\right)^{\alpha}, \end{equation} where $n'$ is a positive integer on interval $[a, \infty)$. Note that (see (\ref{A23})) \begin{equation}\label{A50} A_i^{\alpha}\sim \left(i^s f(i)\right)^{\alpha}. \end{equation} Note that the function $x^s f(x)$ is increasing and therefore we have \begin{equation}\label{A51} \sum^{n}_{i=n'}\left(i^s f(i)\right)^{\alpha}=\int^{n}_{n'}x^{s\alpha}f(x)^{\alpha}\ dx+O\left(n^{s\alpha}f(n)^{\alpha}\right). \end{equation} On the other hand (see (\ref{A12})) \begin{equation}\label{A52} \int^{n}_{n'}x^{s\alpha}f(x)^{\alpha}\ dx \sim \frac{n^{s\alpha +1}f(n)^{\alpha}}{s\alpha +1}. \end{equation} Equations (\ref{A49}), (\ref{A51}) and (\ref{A52}) give \begin{equation}\label{A53} 1+2+\cdots+(n'-1)+\sum^{n}_{i=n'}\left(i^s f(i)\right)^{\alpha}\sim \frac{n^{s\alpha +1}f(n)^{\alpha}}{s\alpha +1}\sim \frac{n}{s \alpha +1}A^{\alpha}_{n}. \end{equation} Finally, (\ref{A53}), (\ref{A50}) and Lemma~\ref{L15} give (\ref{A47}). If we substitute $n=\psi (A_n)$ into equation (\ref{A47}) and proceed as in Theorem~\ref{T14} and Theorem~\ref{T16} then we obtain (\ref{A48}). \end{proof} \begin{remark} Equations (\ref{A47}) and (\ref{A48}) when $A_n=p_n$ is the sequence of prime numbers were obtained by S\'alat and Zn\'am \cite {SalatZnam6}, more precise formulas when $\alpha$ is a positive integer were obtained by Jakimczuk \cite {Jakim3}. Equations (\ref{A47}) and (\ref{A48}) when $A_n=c_{n,k}$ is the sequence of numbers with $k$ prime factors were obtained by Jakimczuk \cite {Jakim2}. \end{remark} Jakimczuk \cite {Jakim4} proved the following theorem. \begin{theorem} If $A_n$ satisfies (\ref{A23}) then the following formulas hold \[\sum^{n}_{i=1}\log A_i = s\ n \log n-s\ n + n \log f(n) +o(n),\] \[\lim_{n\rightarrow\infty}\frac{\sqrt[n]{A_1 A_2\ldots A_n}}{A_n}=\frac{1}{e^s}.\] \end{theorem} \begin{proof} See \cite {Jakim4}. In that proof we supposed that \[\lim_{x\rightarrow \infty}\int^{x}_{a}\frac{t f'(t)}{f(t)}\ dt = \infty.\] Consequently (L'Hospital's rule) \begin{equation}\label{A54} \lim_{x\rightarrow \infty}\frac{\int^{x}_{a}\frac{t f'(t)}{f(t)}\ dt}{x} = 0. \end{equation} This supposition is unnecessary since if the integral converges then (\ref{A54}) also holds. \end{proof} \begin{definition} The function of slow increase $f(x)$ is a {\em universal function} if and only if for all sequence $A_n$ that satisfies (\ref{A23}) we have $f(A_n)\sim l f(n)$ where $l$ depends of the sequence $A_n$. \end{definition} \begin{example} Equation (\ref{A28}) implies that $f(x)=\log x$ is an universal function, in this case $l=s$. Equation (\ref{A29}) implies that $f(x)=\log \log x$ is an universal function, in this case $l=1$ does not depend of the sequence $A_n$. \end{example} \begin{remark} Note that if $f(x)$ and $g(x)$ are universal functions then $f(x)^{\alpha}$ $(\alpha>0)$, $Cf(x)$ $(C>0)$ and $f(x)g(x)$ are universal functions. If $f(x)/g(x)$ is a function of slow increase then is an universal function. \end{remark} \begin{theorem} If $f(x)$ is an universal function and $A_n$ satisfies (\ref{A23}) then we have \[\psi(x)\sim \frac{\sum_{A_i\leq x}f(A_i)^{\beta}}{f(x)^{\beta}}\] for all $\beta$. \end{theorem} \begin{proof} The proof is the same as in Theorem~\ref{T16} and Theorem~\ref{T18}. \end{proof} \begin{example} Since $f(x)=\log x $ is an universal function, we have for all sequences $A_n$ satisfying (\ref{A23}) that \[\psi(x)\sim \frac{\sum_{A_i\leq x}\log^{\beta} A_i}{\log^{\beta} x}.\] In particular, if $\beta =1$ we have \[\psi(x)\sim \frac{\sum_{A_i\leq x}\log A_i}{\log x}.\] \end{example} \begin{theorem} There exist functions of slow increase that are not universal functions. \end{theorem} \begin{proof} We shall prove that the following function of slow increase \[g(x)=e^{\frac{\log x}{\log\log x}},\] is not an universal function. We shall prove that there exists a sequence $A_n$ that satisfies (\ref{A23}) and \[\lim_{n\rightarrow \infty}\frac{g(A_n)}{g(n)}= \infty.\] Since $A_n$ satisfies (\ref{A23}) we can write \[A_n=h_1(n) n^s f(n),\] where $h_1(n)\rightarrow 1$. Therefore \begin{equation}\label{A55} \frac{g(A_n)}{g(n)}=\exp\left(\frac{\log h_1(n)+s \log n+\log f(n)}{\log\log n+\log s+\log \left(1+\frac{\log f(n)}{s\log n}+\frac{\log h_1(n)}{s\log n} \right)}-\frac{\log n}{\log\log n}\right). \end{equation} If $s>1$ (\ref{A55}) becomes (see (\ref{A2})) \[\frac{g(A_n)}{g(n)}=\exp\left(h_2(n)\frac{s \log n}{\log\log n}-\frac{\log n}{\log\log n}\right),\] where $h_2(n)\rightarrow 1$. That is \[\frac{g(A_n)}{g(n)}=\exp\left(h_3(n)\frac{(s-1) \log n}{\log\log n}\right),\] where $h_3(n)\rightarrow 1$. Consequently we have \[\lim_{n\rightarrow \infty}\frac{g(A_n)}{g(n)}= \infty.\] This proves the theorem. In particular this limit is true if $f(x)=g(x)$. To complete, we shall examine the case $s=1$. In this case (\ref{A55}) becomes (note that $\lim_{x\rightarrow 0}\frac{\log(1+x)}{x}=1$) \begin{eqnarray} \frac{g(A_n)}{g(n)}&=&\exp\left(\frac{\log h_1(n)+ \log n+\log f(n)}{\log\log n+h_4(n)\frac{\log f(n)}{\log n}+h_4(n)\frac{\log h_1(n)}{\log n} }-\frac{\log n}{\log\log n}\right)\nonumber\\&=& \exp\left(\frac{ \log n+\log f(n)}{\log\log n+h_4(n)\frac{\log f(n)}{\log n}+h_4(n)\frac{\log h_1(n)}{\log n} }-\frac{\log n}{\log\log n}+o(1)\right)\nonumber\\&=& \exp\left(\frac{ \log\log n\log f(n)-h_4(n)\log f(n)-h_4(n)\log h_1(n)}{(\log\log n)^{2} +h_4(n)\frac{\log\log n\log f(n)}{\log n}+h_4(n)\frac{\log\log n\log h_1(n)}{\log n} }+o(1)\right)\nonumber\\&=&\exp\left(h_5(n)\frac{\log f(n)}{\log\log n}+o(1)\right), \nonumber \end{eqnarray} where $h_4(n)\rightarrow 1$ and $h_5(n)\rightarrow 1$. For example, if $f(x)=g(x)$ then $\lim_{n\rightarrow \infty}\frac{g(A_n)}{g(n)}= \infty$. If $f(x)=\log^{\alpha}x$ $(\alpha>0)$ then $\lim_{n\rightarrow \infty}\frac{g(A_n)}{g(n)}= e^{\alpha}$. If $f(x)=\log\log x$ then $\lim_{n\rightarrow \infty}\frac{g(A_n)}{g(n)}= 1$. \end{proof} \section{Acknowledgements} The author would like to thank the anonymous referees for their valuable comments and suggestions for improving the original version of this manuscript. \begin{thebibliography}{99} \bibitem{Hardy1} G. H. Hardy and E. M. Wright, \textit{An Introduction to the Theory of Numbers}, Fourth Edition, 1960. \bibitem{Jakim2} R. Jakimczuk, A note on sums of powers which have a fixed number of prime factors, \textit{J. Inequal. Pure Appl. Math.} \textbf{6} (2005), Article 31. \bibitem{Jakim3} R. Jakimczuk, Desigualdades y f\'ormulas asint\'oticas para sumas de potencias de primos, \textit{Bol. Soc. Mat. Mexicana} (3) \textbf{11} (2005), 5--10. \bibitem{Jakim4} R. Jakimczuk, The ratio between the average factor in a product and the last factor, \textit{Mathematical Sciences: Quarterly Journal} \textbf{1} (2007), 53--62. \bibitem{ReyPastor5} J. Rey Pastor, P. Pi Calleja, C. Trejo, \textit{An\'alisis Matem\'atico,} Volume I, Editorial Kapeluz, Octava Edici\'on, 1969. \bibitem{SalatZnam6} T. S\'alat and S. Zn\'am, On the sums of prime powers, \textit{Acta. Fac. Rer. Nat. Univ. Com. Math.} \textbf{21} (1968), 21--25. \end{thebibliography} \bigskip \hrule \bigskip \noindent 2000 {\it Mathematics Subject Classification}: Primary 11B99; Secondary 11N45. \noindent \emph{Keywords: } Functions of slow increase, integer sequences, asymptotic formulas. \bigskip \hrule \bigskip \vspace*{+.1in} \noindent Received September 14 2009; revised version received December 21 2009. Published in {\it Journal of Integer Sequences}, December 23 2009. \bigskip \hrule \bigskip \noindent Return to \htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}. \vskip .1in \end{document}