On inversion of absolutely convergent weighted Dirichlet series in two variables

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