Title: Factorisation of the product of Dirichlet series of completely multiplicative functions,
Abstract: In the first chapter, we will present a computation of the square value of
the module of L functions associated to a Dirichlet character. This computation
suggests to ask if a certain ring of arithmetic multiplicative functions exists
and if it is unique. This search has led to the construction of that ring in
chapter two. Finally, in the third chapter, we will present some propositions
associated with this ring. The result below is one of the main results of this
work :
For F and G two completely multiplicative functions, $ s $ a complex number
such as the dirichlet series $ D(F,s) $ and $ D(G,s) $ converge :
$ \forall F,G \in \mathbb{M}_{c} : D(F,s) \times D(G,s) = D(F \times G,2s)
\times D(F \square G,s) $
where the operation $ \square $ is defined in chapter two as the sum of the
previously mentioned ring. Here are some similar versions, with $ s = x+iy $ :
$ \forall F, G \in \mathbb{M}_{c} : ~ D(F,s) \times D(G,\overline{s}) = D(F
\times G,2x) \times D(\frac{F}{\text{Id}_{e}^{iy}} \square
\frac{G}{\text{Id}_{e}^{-iy}}, x) $
$ \forall F, G \in \mathbb{M}_{c} : ~ |D(F,s)|^{2} = D(|F|^{2},2x) \times
D(\frac{F}{\text{Id}_{e}^{iy}} \square \overline{\frac{F}{\text{Id}_{e}^{iy}}},
x) $