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:<math>\sum_{n\le x}(d(n)-(\log n+2\gamma))=o(x)\quad(x\rightarrow\infty),</math>
which suggests that the function <math>\log n+2\gamma</math> is a better choice of average order for <math>d(n)</math> than simply <math>\log n</math>.
==Mean values over {{math|'''F'''<sub>'''q'''</sub>'''[x]'''}}==
===Definition===
Let ''h(x)'' be a function on the set of [[monic polynomial]]s over [[finite field|'''F<sub>q</sub>''']]. For <math>n\ge 1</math> we define
<math>\text{Ave}_{n}(h)=\frac{1}{q^{n}}\sum_{f \text{ monic},\text{ deg}(f)= n}h(f)</math>.
This is the mean value of ''h'' on the set of monic polynomials of degree ''n''. We define the mean value of ''h'' to be
<math>\lim_{n\rightarrow\infty}\text{Ave}_{n}(h)</math> provided this limit exists.
===Zeta function and Dirichlet series in {{math|'''F'''<sub>'''q'''</sub>[X]}}===
Let {{math|'''F<sub>q</sub>[X]'''}}=''A'' be the [[ring of polynomials]] over the [[finite field]] {{math|'''F<sub>q</sub>'''}}.
Let ''h'' be a polynomial arithmetic function (i.e. a function on set of monic polynomials over ''A''). Its corresponding Dirichlet series define to be
<math>D_{h}(s)=\sum_{f\text{ monic}}h(f)|f|^{-s}</math>,
where for <math>g\in A</math>, set <math>|g|=q^{deg(g)}</math> if <math>g\ne 0</math>, and <math>|g|=0</math> otherwise.
The polynomial zeta function is then
<math>\zeta_{A}(s)=\sum_{f\text{ monic}}|f|^{-s}</math>.
Similar to the situation in {{math|'''N'''}}, every Dirichlet series of a [[multiplicative function]] ''h'' has a product representation ([[Euler product]]):
<math>D_{h}(s)=\prod_{P}(\sum_{n\mathop =0}^{\infty}h(P^{n})|P|^{-sn})</math>,
Where the product runs over all monic irreducible polynomials ''P''.
For example, the product representation of zeta function still holds: <math>\zeta_{A}(s)=\prod_{P}(1-|P|^{-s})^{-1}</math>.
Unlike the classical [[zeta function]], <math>\zeta_{A}(s)</math> is very simple:
<math>\zeta_{A}(s)=\sum_{f}(|f|^{-s})=\sum_{n}\sum_{\text{deg(f)=n}}q^{-sn}=\sum_{n}(q^{n-sn})=(1-q^{1-s})^{-1}</math>.
In a similar way, If ''ƒ'' and ''g'' are two polynomial arithmetic functions, one defines ''ƒ'' * ''g'', the ''Dirichlet convolution'' of ''ƒ'' and ''g'', by
:<math>
\begin{align}
(f*g)(m)
&= \sum_{d\,\mid \,m} f(m)g\left(\frac{m}{d}\right) \\
&= \sum_{ab\,=\,f}f(a)g(b)
\end{align}
</math>
where the sum extends over all monic [[divisor]]s ''d'' of ''m'', or equivalently over all pairs (''a'', ''b'') of monic polynomials whose product is ''m''.
The identity <math>D_{h}D_{g}=D_{h*g}</math> still holds. Thus, like in the elementary theory, the polynomial Dirichlet series and the zeta function has a connection with the notion of mean values in the context of polynomials. The following examples illustrate it.
====The density of the k-th power free polynomials in {{math|'''F'''<sub>'''q'''</sub>[X]}}====
Define <math>\delta(f)</math> to be 1 if <math>f</math> is k-th power free and 0 otherwise.
We calculate the average value of δ, which is the density of the k-th power free polynomials in {{math|'''F'''<sub>'''q'''</sub>[X]}}.
By multiplicativity of <math>\delta</math>:
<math>\sum_{f}\frac{\delta(f)}{|f|^{s}}=\prod_{P}(\sum_{j \mathop =0}^{k-1}(|P|^{-js}))=\prod_{P}\frac{1-|P|^{-sk}}{1-|P|^{-s}}=\frac{\zeta_{A}(s)}{\zeta_{A}(sk)}=\frac{1-q^{1-ks}}{1-q^{1-s}}</math>
Denote <math>b_{n}</math> the number of k-th power monic polynomials of degree ''n'', we get
<math>\sum_{f}\frac{\delta(f)}{|f|^{s}}=\sum_{n}\sum_{\text{def}f=n}\delta(f)|f|^{-s}=\sum_{n}b_{n}q^{-sn}</math>.
Making substitution <math>u=q^{-s}</math> we get:
<math>\frac{1-qu^{k}}{1-qu}=\sum_{n \mathop =0}^{\infty}b_{n}u^{n}</math>.
Finally, expand the left-hand side in a geometric series and compare the coefficients on <math>u^{n}</math> on both sides, we get that
<math>b_{n}=\begin{cases}
\;\;\,q^{n} & n\le k-1 \\
\;\;\, q^{n}(1-q^{1-k}) &\text{otherwise} \\
\end{cases}</math>
Hence,
<math>\text{Ave}_{n}(\delta)=1-q^{1-k}=\frac{1}{\zeta_{A}(k)}</math>
And since it doesn't depend on ''n'' this is also the mean value of <math>\delta(f)</math>.
====Number of divisors====
Let <math>d(f)</math> be the number of monic divisors of ''f'' and let <math>D(n)</math> be the sum of <math>d(f)</math> over all monics of degree n.
<math>\zeta_{A}(s)^{2}=(\sum_{h}|h|^{-s})(\sum_{g}|g|^{-s})=\sum_{f}(\sum_{hg=f}1)|f|^{-s}=\sum_{d(f)}|f|^{-s}=D_{d}(s)=\sum_{n \mathop =0}^{\infty}D(n)u^{n}</math>
where <math>u=q^{-s}</math>.
Expanding the right-hand side into power series we get,
<math>D(n)=(n+1)q^{n}</math>.
Substitute <math>x=q^{n}</math> the above equation becomes:
<math>D(n)=xlog_{q}(x)+x</math> which resembles closely the analogous result for integers <math>\sum_{k \mathop =1}^{n}d(k)=xlogx+(2\gamma-1)x+O(\sqrt{x})</math>, where <math>\gamma</math> is [[Euler constant]]. It is a famous problem in [[elementary number theory]] to find the error term. In the polynomials case, there is no error term. This is because of the very simple nature of the zeta function <math>\zeta_{A}(s)</math>.
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