On algebras of Dirichlet series invariant under permutations of coefficients

Let Oᵤ be the algebra of holomorphic functions on C_+: =\s C: Re s>0\ that are limits of Dirichlet series D=₍=₁^ aₙ n^-s, s C_+, that converge uniformly on proper half-planes of C_+. We study algebraic-topological properties of subalgebras of Oᵤ: the Banach algebras W, A, H^ and the Frechet algebra Ob. Here W consists of functions in Oᵤ of absolutely convergent Dirichlet series on the closure of C_+, A is the uniform closure of W, H^ is the algebra of all bounded functions in Oᵤ, and Ob is set of all f (s) =₍=₁^ aₙ n^-s in Oᵤ so that fᵣ H^, r (0, 1), where fᵣ (s): =₍=₁^ aₙ r^ (n) n^-s and (n) is the number of prime factors of n. Let SN be the group of permutations of N. Each SN determines a permutation SN (i. e. , such that (mn) = (n) (m) for all m, n N) via the fundamental theorem of arithmetic. For a Dirichlet series D=₍=₁^ aₙ n^-s, and SN, S_ (D) =₍=₁^ a^-₁ (n) n^-s determines an action of SN 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 SN, 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.