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 $\\epsilon>0$ theset

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)