On inversion of absolutely convergent weighted Dirichlet series in two variables

TL;DR

This paper investigates the inversion of absolutely convergent weighted Dirichlet series in two variables, establishing conditions under which the reciprocal also admits a similar Dirichlet series representation, and explores related composition and weight properties.

Contribution

It provides a characterization of when the reciprocal of a two-variable Dirichlet series has a similar absolutely convergent representation, extending the theory of weighted Dirichlet series in multiple variables.

Findings

Reciprocal series exist if and only if the original is bounded away from zero.

Constructs weights that preserve properties under inversion and composition.

Establishes conditions for the Dirichlet series of composed functions.

Abstract

Let $0<p\leq 1$, and let $\omega:\mathbb N^2 \to [1,\infty)$ be an almost monotone weight. Let $\mathbb H$ be the closed right half plane in the complex plane. Let $\widetilde a$ be a complex valued function on $\mathbb H^2$ such that $\widetilde a(s_1,s_2)=\sum_{(m,n)\in \mathbb N^2}a(m,n)m^{-s_1}n^{-s_2}$ for all $(s_1,s_2)\in \mathbb H^2$ with $\sum_{(m,n)\in \mathbb N^2} |a(m,n)|^p\omega(m,n)<\infty$. If $\widetilde a$ is bounded away from zero on $\mathbb H^2$, then there is an almost monotone weight $\nu$ on $\mathbb N^2$ such that $1\leq \nu\leq \omega$, $\nu$ is constant if and only if $\omega$ is constant, $\nu$ is admissible if and only if $\omega$ is admissible, the reciprocal $\frac{1}{\widetilde a}$ has the Dirichlet representation $\frac{1}{\widetilde a}(s_1,s_2)=\sum_{(m,n)\in \mathbb N^2}b(m,n)m^{-s_1}n^{-s_2}$ for all $(s_1,s_2)\in \mathbb H^2$ and $\sum_{(m,n)\in \mathbb N^2}|b(m,n)|^p\nu(m,n)<\infty$. If $\varphi$ is holomorphic on a neighbourhood of the closure of range of $\widetilde a$, then there is an almost monotone weight $\xi$ on $\mathbb N^2$ such that $1\leq \xi\leq \omega$, $\xi$ is constant if and only if $\omega$ is constant, $\xi$ is admissible if and only if $\omega$ is admissible, $\varphi \circ \widetilde a$ has the Dirichlet series representation $(\varphi\circ \widetilde a)(s_1,s_2)=\sum_{(m,n)\in \mathbb N^2} c(m,n)m^{-s_1}n^{-s_2}\;((s_1,s_2)\in \mathbb H^2)$ and $\sum_{(m,n)\in \mathbb N^2}|c(m,n)|^p\xi(m,n)<\infty$. Let $\omega$ be an admissible weight on $\mathbb N^2$, and let $\widetilde a$ have $p$-th power $\omega$- absolutely convergent Dirichlet series. Then it is shown that the reciprocal of $\widetilde a$ has $p$-th power $\omega$- absolutely convergent Dirichlet series if and only if $\widetilde a$ is bounded away from zero.