THE DE BRUIJN–NEWMAN CONSTANT IS NON-NEGATIVE

Abstract

For each $t\in \mathbb{R}$, we define the entire function $$\begin{eqnarray}H_{t}(z):=\int _{0}^{\infty }e^{tu^{2}}\unicode[STIX]{x1D6F7}(u)\cos (zu)\,du,\end{eqnarray}$$ where $\unicode[STIX]{x1D6F7}$ is the super-exponentially decaying function $$\begin{eqnarray}\unicode[STIX]{x1D6F7}(u):=\mathop{\sum }_{n=1}^{\infty }(2\unicode[STIX]{x1D70B}^{2}n^{4}e^{9u}-3\unicode[STIX]{x1D70B}n^{2}e^{5u})\exp (-\unicode[STIX]{x1D70B}n^{2}e^{4u}).\end{eqnarray}$$ Newman showed that there exists a finite constant $\unicode[STIX]{x1D6EC}$ (the de Bruijn–Newman constant) such that the zeros of $H_{t}$ are all real precisely when $t\geqslant \unicode[STIX]{x1D6EC}$. The Riemann hypothesis is equivalent to the assertion $\unicode[STIX]{x1D6EC}\leqslant 0$, and Newman conjectured the complementary bound $\unicode[STIX]{x1D6EC}\geqslant 0$. In this paper, we establish Newman’s conjecture. The argument proceeds by assuming for contradiction that $\unicode[STIX]{x1D6EC}<0$ and then analyzing the dynamics of zeros of $H_{t}$ (building on the work of Csordas, Smith and Varga) to obtain increasingly strong control on the zeros of $H_{t}$ in the range $\unicode[STIX]{x1D6EC}

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德布鲁因-纽曼常数是非负的

对于每一个$t\in\mathbb｛R｝$，我们定义整个函数$$\boot{eqnarray}H_{t} （z）：=\int _｛0｝^｛\infty｝e ^｛tu ^｛2｝｝\unicode[STIX]｛x1D6F7｝^{2}n^{4}e^{9u}-3\unicode{x1D70B}n^{2}e^｛5u｝）\exp（-\unicode[STIX]{x1D70B}n^{2}e^｛4u｝）。\end｛eqnarray｝$$Newman证明了存在一个有限常数$\unicode[STIX]｛x1D6EC｝$（de Bruijn–Newman常数），使得$H_。黎曼假说等价于断言$\unicode[STIX]｛x1D6EC｝\leqslant 0$，Newman猜想补界$\unicode[STIX]｛x1d6C｝\geqslant 0$。本文建立了Newman猜想。该论点通过假设$\unicode[STIX]｛x1D6EC｝＜0$的矛盾，然后分析$H_，从某种意义上说，它们的局部行为（平均而言）就像它们在算术级数中等距一样，间隙保持接近全球平均间隙大小。但后一种说法与关于黎曼ζ函数零点局部分布的已知结果不一致，例如Montgomery的对相关估计。

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