Editing Multiplicative function (section)

==Multiplicative function over {{math|''F''<sub>''q''</sub>[''X'']}}==
Let {{math|1=''A'' = ''F''<sub>''q''</sub>[''X'']}}, the polynomial ring over the [[finite field]] with ''q'' elements. ''A'' is a [[principal ideal domain]] and therefore ''A'' is a [[unique factorization domain]].

A complex-valued function <math>\lambda</math> on ''A'' is called '''multiplicative''' if <math>\lambda(fg)=\lambda(f)\lambda(g)</math> whenever ''f'' and ''g'' are [[relatively prime]].

===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>
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>

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>

In a similar way, If ''f'' and ''g'' are two polynomial arithmetic functions, one defines ''f''&nbsp;*&nbsp;''g'', the ''Dirichlet convolution'' of ''f'' and ''g'', by
: <math display="block">
\begin{align}
(f*g)(m) 
&= \sum_{d \mid m} f(d)g\left(\frac{m}{d}\right) \\
&= \sum_{ab = m}f(a)g(b),
\end{align}
</math>
where the sum is over all monic [[divisor]]s ''d'' of&nbsp;''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.