Zero-free regions near a line. (English) Zbl 1531.30016
The authors consider entire functions of genus \(1\), i.e., entire functions having the representation
\[
f(z) = z^n e^{a+b z} \prod_{k=1}^\infty \left(1-\frac{z}{z_k} \right) e^{\frac{z}{z_k}},
\]
where \(n\in \mathbb{N}\cup\{0\},\) \(z_k \neq 0,\) and \(\sum_{k=1}^\infty \frac{1}{|z_k|^2} < \infty.\) If \(a, b, z_k \in \mathbb{R},\) then such functions form a subset of the famous Laguerre-Pólya class of entire functions. For an entire function \(f\) of genus 1 the authors denote by \(g(z) := - \frac{d}{dz} \log f(z), \) and for a real parameter \(x\) they consider a function \(g(z+x)= \frac{- n}{z+x} + \sum_{k=0}^\infty a_k(x) z^k. \)
Definition. For a given \(N\in \mathbb{N},\) the authors say that an entire function \(f\) of genus 1 belongs to the \(N\)-th order Laguerre-Pólya class, if \(b\in \mathbb{R}\) and the matrix \[ A_N(x) = \left[ \begin{array}{cccc} a_1(x) & a_2(x) & \ldots & a_N(x) \\ a_2(x) & a_3(x) &\ldots & a_{N+1}(x) \\ \vdots& &\ddots&\vdots \\ a_{N}(x) & a_{N+1}(x) & \ldots & a_{2N-1}(x) \end{array} \right] \] is positive semidefinite.
These classes of functions are nested. We mention that if such \(f\) belongs to the classical Laguerre-Pólya class then for every \(N\in \mathbb{N}\) the matrix \(A_N(x)\) is positive semidefinite.
In the paper the following characteristics of the spacing of roots are in use. For a function \(f\) having the representation introduced above let \(c=0\) if \(f\) has multiple roots, and \[c= \inf_{j\ne k} \big\{ |\operatorname{Re}(z_j - z_k)| : z_j \ne \overline{z_k} \big\}\] otherwise. Let \[d = \inf_j \big\{|\operatorname{Im} z_j|: \operatorname{Im} z_j \ne 0 \big\}\]The authors define the aperture of a function to be \(k = \frac{d}{c}\) (if \(c=0\), then \(k =\infty\)).
The results of the paper are of the following kind.
Theorem 1. Let \(f, f(0) \ne 0,\) be an entire function from the first-order Laguerre-Pólya class. Then \(k \geq \frac{\sqrt{3}}{\pi}\).
Theorem 2. Let \(f, f(0) \ne 0,\) be an entire function from the \(N\)-th order Laguerre-Pólya class. Then \(N =O_{k\to \infty} (k^9)\).
The authors discuss some applications of their results and its connections to the Riemann hypothesis.
Definition. For a given \(N\in \mathbb{N},\) the authors say that an entire function \(f\) of genus 1 belongs to the \(N\)-th order Laguerre-Pólya class, if \(b\in \mathbb{R}\) and the matrix \[ A_N(x) = \left[ \begin{array}{cccc} a_1(x) & a_2(x) & \ldots & a_N(x) \\ a_2(x) & a_3(x) &\ldots & a_{N+1}(x) \\ \vdots& &\ddots&\vdots \\ a_{N}(x) & a_{N+1}(x) & \ldots & a_{2N-1}(x) \end{array} \right] \] is positive semidefinite.
These classes of functions are nested. We mention that if such \(f\) belongs to the classical Laguerre-Pólya class then for every \(N\in \mathbb{N}\) the matrix \(A_N(x)\) is positive semidefinite.
In the paper the following characteristics of the spacing of roots are in use. For a function \(f\) having the representation introduced above let \(c=0\) if \(f\) has multiple roots, and \[c= \inf_{j\ne k} \big\{ |\operatorname{Re}(z_j - z_k)| : z_j \ne \overline{z_k} \big\}\] otherwise. Let \[d = \inf_j \big\{|\operatorname{Im} z_j|: \operatorname{Im} z_j \ne 0 \big\}\]The authors define the aperture of a function to be \(k = \frac{d}{c}\) (if \(c=0\), then \(k =\infty\)).
The results of the paper are of the following kind.
Theorem 1. Let \(f, f(0) \ne 0,\) be an entire function from the first-order Laguerre-Pólya class. Then \(k \geq \frac{\sqrt{3}}{\pi}\).
Theorem 2. Let \(f, f(0) \ne 0,\) be an entire function from the \(N\)-th order Laguerre-Pólya class. Then \(N =O_{k\to \infty} (k^9)\).
The authors discuss some applications of their results and its connections to the Riemann hypothesis.
Reviewer: Anna Vishnyakova (Kharkiv)
MSC:
| 30D15 | Special classes of entire functions of one complex variable and growth estimates |
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