The set of all arithmetic functions $f:\mathbb{Z}^{+}\to\mathbb{C}$, under pointwise addition and Dirichlet convolution, is a commutative ring, not all functions are Dirichlet invertible.
So my question is: To what subset of the arithmetic functions and under what norms can one attach a Banach algebra structure to the Dirichlet ring? Is it even possible?
On
If you restrict to the convergent Dirichlet series (those with polynomially bounded coefficients) then $$\|a\|=4\sup_n |a_n| e^{-n}$$ should be a ring norm, as for some $n$
$$\|a\ast b\|=4 |\sum_{d| n}a_d b_{n/d}|e^{-n} \le 4 \sum_{d|n} \frac1{4} \|a\| e^{n/d}\frac1{4} \|b\| e^d e^{-n}\le \frac1{4} \|a\|\|b\|(1 + 2 e^{-n}\sum_{d=2}^{\sqrt n} e^{n/d}e^{d})$$ $$\le \frac1{4} \|a\|\|b\| (1 + 2 \sum_{d=2}^{\sqrt n} e^{d-n/2})\le \|a\|\|b\| $$
The completion is clearly the Banach algebra of formal Dirichlet series with $O(e^n)$ coefficients. It is an integral domain.
Note that the inverse of a convergent Dirichlet series $k^{-s}+\sum_{n=k+1}^\infty c_n n^{-s}$ exists as a convergent generalized Dirichlet series $k^s \sum_{l=0}^\infty (-\sum_{n=k+1}^\infty c_n (n/k)^{-s})^l=\sum_{u\in \Bbb{Q}_{> 0}} d_u u^{-s}$, the obtained ring is a field. Does anyone want to check how $\|a\ast b\|\le \|a\|\|b\|$ fails when extending the norm to those generalized Dirichlet series ?
In fact $$\|a\| = \sum_{n=1}^\infty |a_n n^{-\sigma}|$$ is a bit more natural than my other answer.. It gives the Banach algebra of absolutely convergent Dirichlet series on $\Re(s)\ge \sigma$. There are interesting theorems about its invertible elements.