Refactorisation of the Dirichlet convolution

TL;DR

This paper introduces a novel factorization of the Dirichlet convolution for completely multiplicative functions, leading to a new ring structure and a generalization of Hardy's formula involving the Riemann zeta function.

Contribution

It presents a new factorization method for Dirichlet convolution and constructs an associated ring, extending classical identities like Hardy's formula.

Findings

Derived a generalized Hardy formula involving complex zeta functions

Constructed a new algebraic ring from Dirichlet convolution operations

Provided identities linking multiplicative functions and zeta function properties

Abstract

We present a new way to factor the dirichlet convolution for completely multiplicative functions whitch led us to constructing a ring that arise from the operations involved in the factorisation. We will conclude by some identities that was found during this work. An application of the results gives us a generalisation of the following Hardy formula: $$\zeta(x)^{2} = \zeta(2x)\sum_{m=1}^{+\infty} \frac{2^{\omega(m)}}{m^{x}}$$ which is: $$|\zeta(z)|^{2} = \zeta(2x)\sum_{m=1}^{+\infty}\frac{1}{m^{x}}2^{\omega(m)}\prod_{p | m , p \in \mathbb{P}}^{\omega(m)}\cos(y\ln(p^{v_{p}(m)}))$$ with: $z$ a complex number with $z = x+iy$ and $\Re(z) > 1 $ and x > 1 in Hardy's formula, $\omega(m)$ number of unique primes in $m$, $v_{p}(m$ power of the prime $p$ in $m$.