normal order

Let $f(n)$ and $F(n)$ be functions from $\mathbb{Z}^{+}\rightarrow\mathbb{R}$ . We say that $f(n)$ has normal order $F(n)$ if for each $\epsilon>0$ the set

A(\epsilon)=\{n\in\mathbb{Z}^{+}:(1-\epsilon)F(n)<f(n)<(1+\epsilon)F(n)\}

has the property that $\underline{d}(A(\epsilon))=1$ . Equivalently, if $B(\epsilon)=\mathbb{Z}^{+}\backslash A(\epsilon)$ , then $\underline{d}(B(\epsilon))=0$ . (Note that $\underline{d}(X)$ denotes the lower asymptotic density of $X$ ).

We say that $f$ has average order $F$ if

\sum_{j=1}^{n}f(j)\sim\sum_{j=1}^{n}F(j)

Title	normal order
Canonical name	NormalOrder
Date of creation	2013-03-22 12:36:23
Last modified on	2013-03-22 12:36:23
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	5
Author	mathcam (2727)
Entry type	Definition
Classification	msc 11B05
Defines	average order