On algebras of Dirichlet series invariant under permutations of coefficients

Abstract

Let $\mathscr O_u$ be the algebra of holomorphic functions on ${\bf C}_+:=\{s\in{\bf C}:\text{Re }s>0\}$ that are limits of Dirichlet series $D=\sum_{n=1}^\infty a_n n^{-s}$, $s\in \bf{C}_+$, that converge uniformly on proper half-planes of $\bf{C}_+$. We study algebraic-topological properties of subalgebras of $\mathscr O_u$: the Banach algebras $\mathscr W, \mathscr A, \mathscr H^\infty$ and the Frechet algebra $\mathscr O_b$. Here $\mathscr W$ consists of functions in $\mathscr O_u$ of absolutely convergent Dirichlet series on the closure of $\bf{C}_+$, $\mathscr A$ is the uniform closure of $\mathscr W$, $\mathscr H^\infty$ is the algebra of all bounded functions in $\mathscr O_u$, and $\mathscr O_b$ is set of all $f(s)=\sum_{n=1}^\infty a_n n^{-s}$ in $\mathscr O_u$ so that $f_r\in \mathscr H^\infty$, $r\in (0,1)$, where $f_r(s):=\sum_{n=1}^\infty a_n r^{\Omega(n)} n^{-s}$ and $\Omega(n)$ is the number of prime factors of $n$. Let $S_\bf{N}$ be the group of permutations of $\bf{N}$. Each $\sigma\in S_\bf{N}$ determines a permutation $\hat\sigma\in S_\bf{N}$ (i.e., such that $\hat\sigma(mn)=\hat\sigma(n)\hat\sigma(m)$ for all $m,n\in \bf{N}$) via the fundamental theorem of arithmetic. For a Dirichlet series $D=\sum_{n=1}^\infty a_n n^{-s}$, and $\sigma \in S_\bf{N}$, $S_\sigma(D)=\sum_{n=1}^\infty a_{\hat{\sigma}^{-1}(n)} n^{-s}$ determines an action of $S_\bf{N}$ on the set of all Dirichlet series. It is shown that each of the algebras above is invariant with respect to this action. Given a subgroup $G$ of $S_\bf{N}$, the set of $G$-invariant subalgebras of these algebras are studied, and their maximal ideal spaces are described, and used to characterise groups of units and of invertible elements having logarithms, find the stable rank, show projective freeness, and describe when the special linear group is generated by elementary matrices, with bounds on the number of factors.