Editing Multiplicative function (section)

== Convolution ==

If ''f'' and ''g'' are two multiplicative functions, one defines a new multiplicative function <math>f * g</math>, the ''[[Dirichlet convolution]]'' of ''f'' and ''g'', by
<math display="block"> (f \, * \, g)(n) = \sum_{d|n} f(d) \, g \left( \frac{n}{d} \right)</math>
where the sum extends over all positive divisors ''d'' of ''n''. 
With this operation, the set of all multiplicative functions turns into an [[abelian group]]; the [[identity element]] is ''ε''. Convolution is commutative, associative, and distributive over addition.

Relations among the multiplicative functions discussed above include:

* <math>\mu * 1 = \varepsilon</math> (the [[Möbius inversion formula]])
* <math>(\mu \operatorname{Id}_k) * \operatorname{Id}_k = \varepsilon</math> (generalized Möbius inversion)
* <math>\varphi * 1 = \operatorname{Id}</math>
* <math>d = 1 * 1</math>
* <math>\sigma = \operatorname{Id} * 1 = \varphi * d</math>
* <math>\sigma_k = \operatorname{Id}_k * 1</math>
* <math>\operatorname{Id} = \varphi * 1 = \sigma * \mu</math>
* <math>\operatorname{Id}_k = \sigma_k * \mu</math>

The Dirichlet convolution can be defined for general arithmetic functions, and yields a ring structure, the [[Dirichlet ring]]. 

The [[Dirichlet convolution]] of two multiplicative functions is again multiplicative. A proof of this fact is given by the following expansion for relatively prime <math>a,b \in \mathbb{Z}^{+}</math>: 
<math display="block">\begin{align}
(f \ast g)(ab)
& = \sum_{d|ab} f(d) g\left(\frac{ab}{d}\right) \\
&= \sum_{d_1|a} \sum_{d_2|b} f(d_1d_2) g\left(\frac{ab}{d_1d_2}\right) \\
&= \sum_{d_1|a} f(d_1) g\left(\frac{a}{d_1}\right) \times \sum_{d_2|b} f(d_2) g\left(\frac{b}{d_2}\right) \\
&= (f \ast g)(a) \cdot (f \ast g)(b).
\end{align} </math>

=== Dirichlet series for some multiplicative functions ===
* <math>\sum_{n\ge 1} \frac{\mu(n)}{n^s} = \frac{1}{\zeta(s)}</math>
* <math>\sum_{n\ge 1} \frac{\varphi(n)}{n^s} = \frac{\zeta(s-1)}{\zeta(s)}</math>
* <math>\sum_{n\ge 1} \frac{d(n)^2}{n^s} = \frac{\zeta(s)^4}{\zeta(2s)}</math>
* <math>\sum_{n\ge 1} \frac{2^{\omega(n)}}{n^s} = \frac{\zeta(s)^2}{\zeta(2s)}</math>
More examples are shown in the article on [[Dirichlet series]].