normal order

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

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

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

$$\sum _{j=1}^{n}f(j)\sim \sum _{j=1}^{n}F(j)$$