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G. Pólya proved that given a sequence of positive real numbers Λ = {λn}n=1 of a density d and satisfying the gap condition infn∈N(λn+1 − λn) > 0, the Dirichlet series ∑∞ n=1 cne 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

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