Article
Ufa Mathematical Journal
Volume 10, Number 3, pp. 142-148
A Taylor-Dirichlet series with no singularities on its abscissa of convergence
Zikkos E.
DOI:10.13108/2018-10-3-142
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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:
\noindent
(1) $\Lambda$ has density $d$, that is, $\sum_{\lambda_n\le t}\mu_n/t\to d$ as $t\to\infty$,
\noindent
(2) $\lambda_n$ 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.