# 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)

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 NormalOrder 2013-03-22 12:36:23 2013-03-22 12:36:23 mathcam (2727) mathcam (2727) 5 mathcam (2727) Definition msc 11B05 average order