On the entire functions from the Laguerre-Pólya I class having the increasing second quotients of Taylor coefficients

On the closest to zero roots and the second quotients of Taylor coefficients of entire functions from the Laguerre-Pólya I class

For an entire function $f(z) = \sum_{k=0}^\infty a_k z^k, a_k>0,$ we show that if $f$ belongs to the Laguerre-Pólya class, and the quotients $q_k := \frac{a_{k-1}^2}{a_{k-2}a_k}, k=2, 3, \ldots $ satisfy the condition $q_2 \leq q_3,$ then $f$ has at least one zero in the segment $[-\frac{a_1}{a_2},0].$ We also give necessary conditions and sufficient conditions of the existence of such a zero in terms of the quotients $q_k$ for $k=2,3, 4.$

On the entire functions from the Laguerre-Pólya I class with non-monotonic second quotients of Taylor coefficients

We study the entire functions $f(z) = \sum_{k=0}^\infty a_k z^k, a_k>0,$ with non-monotonic second quotients of Taylor coefficients, namely, such that $\frac{a_{2m-1}^2}{a_{2m-2}a_{2m}} = a>1$ and $\frac{a_{2m}^2}{a_{2m-1}a_{2m+1}} = b>1$ for all $m \in \mathbb{N}.$ We obtain necessary and sufficient conditions under which such functions belong to the Laguerre-Pólya I class.

On the necessary condition for entire function with the increasing second quotients of Taylor coefficients to belong to the Laguerre-Pólya class

For an entire function $f(z) = \sum_{k=0}^\infty a_k z^k, a_k>0,$ we show that $f$ does not belong to the Laguerre-Pólya class if the quotients $\frac{a_{n-1}^2}{a_{n-2}a_n}$ are increasing in $n$, and $c:= \lim\limits_{n\to \infty} \frac{a_{n-1}^2}{a_{n-2}a_n}$ is smaller than an absolute constant $q_\infty$ $(q_\infty\approx 3{.}2336) .$

On the number of real zeros of real entire functions with a non-decreasing sequence of the second quotients of Taylor coefficients

For an entire function $f(z) = \sum_{k=0}^\infty a_k z^k,$ $a_k >0,$ we define the sequence of the second quotients of Taylor coefficients $Q := \left( \frac{a_k^2}{a_{k-1}a_{k+1}} \right)_{k=1}^\infty$. We find new necessary conditions for a function with a non-decreasing sequence $Q$ to belong to the Laguerre--Pólya class of type I. We also estimate the possible number of nonreal zeros for a function with a non-decreasing sequence $Q.$

Riemann xi-函数泰勒系数的无限对数凹性

Infinite log-concavity of the Taylor coefficients of the Riemann xi-function

本文证明了Riemann xi-函数泰勒系数序列严格无限对数凹，解决了Zhu的猜想，方法结合复解析估计、区间算术与全局闭包论证。