A Taylor-Dirichlet series with no singularities on its abscissa of convergence

G. Pólya proved that given a sequence of positive real numbers Λ = [λ_n]_n=1 ^∞ of a density d and satisfying the gap gap condition inf n ∈ ℕ(λ_n+1-λ_n) > 0, the Dirichlet series Σ _n=1 ^∞ c_ne^λnz has at least one singularity in each open interval whose length exceeds 2πd and lies on the abscissa of convergence. This raises the question whether the same result holds for a Taylor-Dirichlet series of the form when its associated multiplicity-sequence Λ = [λ_n, μ_n] _n=1 ^∞.