Institute of Mathematics

Abstract:  Let $0<p\leq1$, 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 (nonconstant) if $\omega$ is constant (nonconstant), $\nu$ is admissible (nonadmissible) if $\omega$ is admissible (nonadmissible), the reciprocal $\frc1{\widetilde a}$ has the Dirichlet representation \frac1{\widetilde a}(s_1,s_2)=\sum_{(m,n)\in\mathbb N^2}b(m,n)m^{-s_1}n^{-s_2} \quad\text{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 (nonconstant) if $\omega$ is constant (nonconstant), $\xi$ is admissible (nonadmissible) if $\omega$ is admissible (nonadmissible), $\varphi\circ\widetilde a$ 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 a $p$th power $\omega$-absolutely convergent Dirichlet series. Then it is shown that the reciprocal of $\widetilde a$ has a $p$th power $\omega$-absolutely convergent Dirichlet series if and only if $\widetilde a$ is bounded away from zero. Though proofs are given for a function of two variables, it can naturally be extended for a function of several variables.