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This article is cited in 1 scientific paper (total in 1 paper)
A Taylor–Dirichlet series with no singularities on its abscissa of convergence
E. Zikkos Department of Mathematics and Statistics,
POB 20537, University of Cyprus,
1678 Nicosia, Cyprus
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.
Keywords:
Taylor–Dirichlet series, singularities, Fabry–Pólya.
Received: 30.05.2017
Citation:
E. Zikkos, “A Taylor–Dirichlet series with no singularities on its abscissa of convergence”, Ufimsk. Mat. Zh., 10:3 (2018), 146–152; Ufa Math. J., 10:3 (2018), 142–148
Linking options:
https://www.mathnet.ru/eng/ufa442https://doi.org/10.13108/2018-10-3-142 https://www.mathnet.ru/eng/ufa/v10/i3/p146
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Abstract page: | 168 | Russian version PDF: | 54 | English version PDF: | 8 | References: | 19 |
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