Lagrange Inversion Theorem for Dirichlet series

TL;DR

This paper extends the classical Lagrange Inversion Theorem to Dirichlet series, using properties of Dirichlet convolution polynomials to establish the analogue.

Contribution

It introduces a novel Lagrange Inversion Theorem for Dirichlet series, expanding the mathematical toolkit for analyzing these functions.

Findings

Established an inversion formula for Dirichlet series.

Connected Dirichlet convolution polynomials to classical inversion techniques.

Provided a new approach for studying Dirichlet series properties.

Abstract

We prove an analogue of the Lagrange Inversion Theorem for Dirichlet series. The proof is based on studying properties of Dirichlet convolution polynomials, which are analogues of convolution polynomials introduced by Knuth in [4].