Exercise 2, Appendix 0

Let D be the total algebra of the monoid $\mathbf{N}^*$ (Algebra, Chapter III, § 2, no. 10). If $n \in \mathbf{N}^*$, let $n^\omega$ denote its image in D, so that $1^\omega = 1$ and $(nm)^\omega = n^\omega m^\omega$ for $n, m$ in $\mathbf{N}^*$.† Every element $f$ of D can be written uniquely as a series $\sum_{n=1}^\infty a_n n^\omega$, where $a_n \in K$.

(a) Let $f = \sum_{n=1}^\infty a_n n^\omega$ be an element of D. Show that $f$ is invertible in D if and only if $a_1$ is invertible in K. In particular, if K is a local ring, so is D.

(b) We write $\zeta = \sum_{n=1}^\infty n^\omega$ and $\mu = \sum_{n=1}^\infty \mu(n) n^\omega$. Show that $\zeta$ and $\mu$ are inverses of one another. Deduce that, if $s = \sum_n s(n) n^\omega$ and $t = \sum_n t(n) n^\omega$ are two elements of D, the relations $s = \zeta \cdot t$ and $t = \mu \cdot s$ are equivalent (variant of the Möbius inversion formula).

(c) Let P be the set of prime numbers. Show that the family of $(1 - p^\omega)$ for $p \in P$ is multipliable in D and that

† We often write $-s$ instead of $\omega$; we then say that the elements of D are the Dirichlet formal power series with coefficients in K.