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===Zeta function and Dirichlet series in {{math|''F''<sub>''q''</sub>[''X'']}}===
Let ''h'' be a polynomial arithmetic function (i.e. a function on set of monic polynomials over ''A''). Its corresponding Dirichlet series is defined to be
: <math display="block">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 display="block">\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 display="block">D_{h}(s)=\prod_P \left(\sum_{n\mathop =0}^{\infty}h(P^{n})|P|^{-sn}\right),</math>▼
▲Similar to the situation in {{math|'''N'''}}, every Dirichlet series of a multiplicative function ''h'' has a product representation ([[Euler product]]):
▲: <math display="block">D_{h}(s)=\prod_P \left(\sum_{n\mathop =0}^{\infty}h(P^{n})|P|^{-sn}\right),</math>
where the product runs over all monic irreducible polynomials ''P''. For example, the product representation of the zeta function is as for the integers:
: <math display="block">\zeta_A(s)=\prod_{P}(1-|P|^{-s})^{-1}.</math>▼
▲<math display="block">\zeta_A(s)=\prod_{P}(1-|P|^{-s})^{-1}.</math>
Unlike the classical [[zeta function]], <math>\zeta_A(s)</math> is a simple rational function:
: <math display="block">\zeta_A(s)=\sum_f |f|^{-s} = \sum_n\sum_{\deg(f)=n}q^{-sn}=\sum_n(q^{n-sn})=(1-q^{1-s})^{-1}.</math>▼
▲<math display="block">\zeta_A(s)=\sum_f |f|^{-s} = \sum_n\sum_{\deg(f)=n}q^{-sn}=\sum_n(q^{n-sn})=(1-q^{1-s})^{-1}.</math>
In a similar way, If ''f'' and ''g'' are two polynomial arithmetic functions, one defines ''f'' * ''g'', the ''Dirichlet convolution'' of ''f'' and ''g'', by
: <math display="block">▼
▲<math display="block">
\begin{align}
(f*g)(m)
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\end{align}
</math>
where the sum is 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.
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