Taylor coefficients and zeroes of entire functions of exponential type
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| Format: | Preprint |
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2025
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| _version_ | 1866915972862967808 |
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| author | Hadassi, Lior Sodin, Mikhail |
| author_facet | Hadassi, Lior Sodin, Mikhail |
| contents | Let $F$ be an entire function of exponential type represented by the Taylor series \[ F(z) = \sum_{n\ge 0} ω_n \frac{z^n}{n!} \] with unimodular coefficients $|ω_n|=1$. We show that either the counting function $n_F(r)$ of zeroes of $F$ grows linearly at infinity, or $F$ is an exponential function. The same conclusion holds if only a positive asymptotic proportion of the coefficients $ω_n$ is unimodular. This significantly extends a classical result of Carlson (1915).
The second result requires less from the coefficient sequence $ω$, but more from the counting function of zeroes $n_F$. Assuming that $0<c\le |ω_n| \le C <\infty$, $n\in\mathbb Z_+$, we show that $n_F(r) = o(\sqrt{r})$ as $r\to\infty$, implies that $F$ is an exponential function. The same conclusion holds if, for some $α<1/2$, $n_F(r_j)=O(r_j^α)$ only along a sequence $r_j\to\infty$. Furthermore, this conclusion ceases to hold if $n_F(r)=O(\sqrt r)$ as $r\to\infty$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_13104 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Taylor coefficients and zeroes of entire functions of exponential type Hadassi, Lior Sodin, Mikhail Complex Variables Let $F$ be an entire function of exponential type represented by the Taylor series \[ F(z) = \sum_{n\ge 0} ω_n \frac{z^n}{n!} \] with unimodular coefficients $|ω_n|=1$. We show that either the counting function $n_F(r)$ of zeroes of $F$ grows linearly at infinity, or $F$ is an exponential function. The same conclusion holds if only a positive asymptotic proportion of the coefficients $ω_n$ is unimodular. This significantly extends a classical result of Carlson (1915). The second result requires less from the coefficient sequence $ω$, but more from the counting function of zeroes $n_F$. Assuming that $0<c\le |ω_n| \le C <\infty$, $n\in\mathbb Z_+$, we show that $n_F(r) = o(\sqrt{r})$ as $r\to\infty$, implies that $F$ is an exponential function. The same conclusion holds if, for some $α<1/2$, $n_F(r_j)=O(r_j^α)$ only along a sequence $r_j\to\infty$. Furthermore, this conclusion ceases to hold if $n_F(r)=O(\sqrt r)$ as $r\to\infty$. |
| title | Taylor coefficients and zeroes of entire functions of exponential type |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2504.13104 |