# Factorisation of the product of Dirichlet series of completely multiplicative functions

## TL;DR

This paper explores the factorization properties of Dirichlet series of completely multiplicative functions, constructing a unique ring of such functions and establishing new identities involving their products and associated Dirichlet series.

## Contribution

It introduces a novel ring structure for arithmetic multiplicative functions and derives new factorization formulas for the product of their Dirichlet series.

## Key findings

- Derived a key factorization formula for Dirichlet series of multiplicative functions.
- Constructed a unique ring of multiplicative functions based on these factorizations.
- Established identities relating the modulus squared of Dirichlet series to products within the ring.

## 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) $