Inverse Problem for a Monoid with an Exponential Sequence of Primes

In this paper, for an arbitrary monoid of natural numbers, the foundations of the Dirichlet series algebra are constructed over either a number field or the ring of integers of an algebraic number field. For any number field , it is shown that the set of all invertible Dirichlet series of is an infi...

Full description

Saved in:
Bibliographic Details
Published in:Doklady. Mathematics 2022-12, Vol.106 (Suppl 2), p.S181-S191
Main Authors: Dobrovol’skii, N. N., Rebrova, I. Yu, Dobrovol’skii, N. M.
Format: Article
Language:English
Subjects:
Algebra
Convergence
Dirichlet problem
Domains
Fields (mathematics)
Group theory
Half planes
Integers
Inverse problems
Mathematical analysis
Mathematics
Mathematics and Statistics
Monoids
Number theory
Rings (mathematics)
Theorems
Citations: Items that this one cites
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:In this paper, for an arbitrary monoid of natural numbers, the foundations of the Dirichlet series algebra are constructed over either a number field or the ring of integers of an algebraic number field. For any number field , it is shown that the set of all invertible Dirichlet series of is an infinite Abelian group consisting of series whose first coefficient is nonzero. We introduce the notion of an entire Dirichlet series of a monoid of natural numbers that form an algebra over the ring of algebraic integers of . For the group of algebraic units of the ring of algebraic integers of , it is shown that the set of entire Dirichlet series with is a multiplicative group. For any Dirichlet series from the Dirichlet series algebra of a monoid of natural numbers, the reduced series, the noninvertible part, and the additional series are defined. A formula for decomposition of an arbitrary Dirichlet series into the product of a reduced series and a construction of a noninvertible part and an additional series is found. For any monoid of natural numbers, the algebra of Dirichlet series convergent in the whole complex domain is defined. The Dirichlet series algebra with a given half-plane of absolute convergence is constructed. It is shown that, for any nontrivial monoid M and any real , there is an infinite set of Dirichlet series from such that the domain of their holomorphy is the α-half-plane . With the help of the universality theorem of S.M. Voronin, a weak form of the universality theorem is proved for a wide class of zeta functions of monoids of natural numbers. In conclusion, topical problems with zeta functions of monoids of natural numbers that require further research are described. In particular, if the Linnik–Ibragimov conjecture is true, then the strong universality theorem should be valid for them.
ISSN:1064-5624
1531-8362
DOI:10.1134/S1064562422700211