On algebras of Dirichlet series invariant under permutations of coefficients

2024/04/04 by Alexander Brudnyi, Brudnyi, Alexander, Amol Sasane +1 · 2 citations
Mathematics · #32A10 #46G20 #46J15 #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Primary 30B50 #Rings and Algebras (math.RA) #Secondary 32A05 #advanced mathematical theories

paper · doi:10.48550/arxiv.2404.03616

openalex publication_date 2024/04/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \mathscr O_u be the algebra of holomorphic functions on \bf C₊:=\s∈\bf C:Re s>0\ that are limits of Dirichlet series D=∑_n=1^∞ a_n n^-s, s∈ \bfC₊, that converge uniformly on proper half-planes of \bfC₊. We study algebraic-topological properties of subalgebras of \mathscr O_u: the Banach algebras \mathscr W, \mathscr A, \mathscr H^∞ 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 \bfC₊, \mathscr A is the uniform closure of \mathscr W, \mathscr H^∞ is the algebra of all bounded functions in \mathscr O_u, and \mathscr O_b is set of all f(s)=∑_n=1^∞ a_n n^-s in \mathscr O_u so that f_r∈ \mathscr H^∞, r∈ (0,1), where f_r(s):=∑_n=1^∞ a_n r^Ω(n) n^-s and Ω(n) is the number of prime factors of n. Let S_\bfN be the group of permutations of \bfN. Each σ∈ S_\bfN determines a permutation σ∈ S_\bfN (i.e., such that σ(mn)=σ(n)σ(m) for all m,n∈ \bfN) via the fundamental theorem of arithmetic. For a Dirichlet series D=∑_n=1^∞ a_n n^-s, and σ∈ S_\bfN, S_σ(D)=∑_n=1^∞ a_σ^-1(n) n^-s determines an action of S_\bfN 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_\bfN, 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.

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