On inversion of absolutely convergent weighted Dirichlet series in two variables

Let 0<p≤ 1, and let ω: N² → [1,∞) be an almost monotone weight. Let H be the closed right half plane in the complex plane. Let a be a complex valued function on H² such that a(s₁,s₂)=∑(m,n)∈ N²a(m,n)m-s₁n-s₂ for all (s₁,s₂)∈ H² with ∑(m,n)∈ N² |a(m,n)|ᵖω(m,n)<∞. If a is bounded away from zero on H², then there is an almost monotone weight ν on N² such that 1≤ ν≤ ω, ν is constant if and only if ω is constant, ν is admissible if and only if ω is admissible, the reciprocal 1/ a has the Dirichlet representation 1/ a(s₁,s₂)=∑(m,n)∈ N²b(m,n)m-s₁n-s₂ for all (s₁,s₂)∈ H² and ∑(m,n)∈ N²|b(m,n)|ᵖν(m,n)<∞. If φ is holomorphic on a neighbourhood of the closure of range of a, then there is an almost monotone weight ξ on N² such that 1≤ ξ≤ ω, ξ is constant if and only if ω is constant, ξ is admissible if and only if ω is admissible, φ ∘ a has the Dirichlet series representation (φ∘ a)(s₁,s₂)=∑(m,n)∈ N² c(m,n)m-s₁n-s₂\;((s₁,s₂)∈ H²) and ∑(m,n)∈ N²|c(m,n)|ᵖξ(m,n)<∞. Let ω be an admissible weight on N², and let a have p-th power ω- absolutely convergent Dirichlet series. Then it is shown that the reciprocal of a has p-th power ω- absolutely convergent Dirichlet series if and only if a is bounded away from zero.