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And since it doesn't depend on ''n'' this is also the mean value of <math>\delta(f)</math>.
===Examples===
====Polynomial Divisor function====
At {{math|'''F<sub>q</sub>[X]'''}}, we define
<math>\sigma _{k}(m)=\sum_{f|m \text{, f monic}}|f|^{k}</math>.
We will compute <math>\text{Ave}_{n}(\sigma_{k})</math> for <math>k\ge 1</math>.
First, notice that:</br>
<math>\sigma_{k}(m)=h*\mathbb{I}(m)</math>
where <math>h(f)=|f|^{k}</math> and <math>\;\mathbb{I}(f)=1\;\; \forall{f}</math>.
Therefore,
<math>\sum_{m}\sigma_{k}(m)|m|^{-s}=\zeta_{A}(s)\sum_{m}h(m)|m|^{-s}</math>.
Substitute <math>q^{-s}=u</math> we get,
<math>\text{LHS}=\sum_{n}(\sum_{\text{deg}m=n})u^{n}</math>, and by [[Cauchy product]] we get,
<math>\text{RHS}=\sum_{n}(q^{n}(\frac{1-q^{k(n+1)}}{1-q^{k}}))u^{n}</math>.
Finally we get that,
<math> \text{Ave}_{n}\sigma_{k}=\frac{1-q^{k(n+1)}}{1-q^{k}}</math>.
====Number of divisors====
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<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>.
==See also==
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