Refactorisation of the Dirichlet convolution

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: $$\zeta (x{)}^2=\zeta (2x)\sum _{m=1}^{+\mathrm{\infty }}\frac{2^{\omega (m)}}{m^x}$$ which is: $$\mathrm{\mid }\zeta (z){\mathrm{\mid }}^2=\zeta (2x)\sum _{m=1}^{+\mathrm{\infty }}\frac{1}{m^x}{2}^{\omega (m)}\prod _{p\mathrm{\mid }m,p\in \mathbb{P}}^{\omega (m)}\cos (y\ln (p^{v_p(m)}))$$ with: $z$ a complex number with $z=x+iy$ and $\mathrm{\Re }(z)>1$ and x > 1 in Hardy’s formula, $\omega (m)$ number of unique primes in $m$, $v_p(m$ power of the prime $p$ in $m$.