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<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 (
<math>D_{h}(s)=\prod_{P}(\sum_{n\mathop =0}^{\infty}h(P^{n})|P|^{-sn})</math>,
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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.
===Examples===
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