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

Abstract

G. Pólya proved that given a sequence of positive real numbers $\Lambda=\{\lambda_n\}_{n=1}^{\infty}$ of a density $d$ and satisfying the gap condition $\inf_{n\in\mathbb{N}}(\lambda_{n+1}-\lambda_n)>0$, the Dirichlet series $\sum_{n=1}^{\infty}c_ne^{\lambda_n z}$ has at least one singularity in each open interval whose length exceeds $2\pi d$ and lies on the abscissa of convergence. This raises the question whether the same result holds for a Taylor–Dirichlet series of the form $$ g(z)=\sum_{n=1}^{\infty} \left(\sum_{k=0}^{\mu_n-1}c_{n,k} z^k\right) e^{\lambda_n z},\quad c_{n,k}\in \mathbb{C} $$ when its associated multiplicity-sequence $\Lambda=\{\lambda_n,\mu_n\}_{n=1}^{\infty}$ $$ \{\lambda_n,\mu_n\}_{n=1}^{\infty}:=\{\underbrace{\lambda_1,\lambda_1,\dots,\lambda_1}_{\mu_1 - times}, \underbrace{\lambda_2,\lambda_2,\dots,\lambda_2}_{\mu_2 - times},\dots, \underbrace{\lambda_k,\lambda_k,\dots,\lambda_k}_{\mu_k - times},\dots\} $$ has the following two properties: $\Lambda$ has density $d$, that is, $\sum_{\lambda_n\le t}\mu_n/t\to d$ as $t\to\infty$, $\lambda_n$ satisfy the gap condition $\inf_{n\in\mathbb{N}}(\lambda_{n+1}-\lambda_n)>0$. In this article we present a counterexample. We prove that for any non-negative real number $d$ there exists a multiplicity-sequence $\Lambda=\{\lambda_n,\mu_n\}_{n=1}^{\infty}$ having properties (1) and (2), but with unbounded multiplicities $\mu_n$, such that its Krivosheev characteristic $S_{\Lambda}$ is negative. For this $\Lambda$, and based on a result obtained by O.A. Krivosheeva, we show that for any $a\in\mathbb{R}$, there exists a Taylor–Dirichlet series $g(z)$ whose abscissa of convergence is the line $\mathrm{Re}\, z=a$, such that $g(z)$ has no singularities on this line.