Algebra of formal power series, isomorphic to the algebra of formal Dirichlet series

TL;DR

This paper explores an algebra of formal power series isomorphic to the algebra of formal Dirichlet series, introducing matrices similar to Riordan matrices to reveal structural analogies and define a new Riordan-Dirichlet group.

Contribution

It introduces a novel algebraic framework for formal Dirichlet series, including matrices and a group analogous to Riordan matrices and the Riordan group.

Findings

Identified analogies between ordinary power series and Dirichlet series algebras

Defined matrices similar to Riordan matrices for Dirichlet series

Established the Riordan-Dirichlet group and derived identities

Abstract

Ordinary algebra of formal power series in one variable is convenient to study by means of the algebra of Riordan matrices and the Riordan group. In this paper we consider algebra of formal power series without constant term, isomorphic to the algebra of formal Dirichlet series. To study it, we introduce matrices, similar to the Riordan matrices. As a result, some analogies between two algebras becomes visible. For example, the Bell polynomials (polynomials of partitions of number $n$ into $m$ parts) play a certain role in the ordinary algebra. Similar polynomials (polynomials of decompositions of number $n$ into $m$ factors) play a similar role in the considered algebra. Analog of the Lagrange series for the considered algebra is also exists. In connection with this analogy, we introduce matrix group, similar to the Riordan group and called the Riordan-Dirichlet group. As an example, we consider analog of the Abel's identities for this group.