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 $\unicode[STIX]{x1D6EC}<t\leqslant 0$ , until one establishes that the zeros of $H_{0}$ are in local equilibrium, in the sense that they locally behave (on average) as if they were equally spaced in an arithmetic progression, with gaps staying close to the global average gap size. But this latter claim is inconsistent with the known results about the local distribution of zeros of the Riemann zeta function, such as the pair correlation estimates of Montgomery.

中文翻译：

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DE BRUIJN-NEWMAN 常数是非负的

对于每个 $t\in \mathbb{R}$ ，我们定义整个函数 $$\begin{eqnarray}H_{t}(z):=\int _{0}^{\infty }e^{tu^{2}}\unicode[STIX]{x1D6F7}(u)\cos ( zu)\,du,\end{eqnarray}$$ 在哪里 $\unicode[STIX]{x1D6F7}$ 是超指数衰减函数 $$\begin{eqnarray}\unicode[STIX]{x1D6F7}(u):=\mathop{\sum }_{n=1}^{\infty }(2\unicode[STIX]{x1D70B}^{2 }n^{4}e^{9u}-3\unicode[STIX]{x1D70B}n^{2}e^{5u})\exp (-\unicode[STIX]{x1D70B}n^{2}e ^{4u}).\end{eqnarray}$$ 纽曼证明存在一个有限常数 $\unicode[STIX]{x1D6EC}$ （这德布鲁因-纽曼常数) 使得 $H_{t}$ 都是真实的 $t\geqslant \unicode[STIX]{x1D6EC}$ . 黎曼假设等价于断言 $\unicode[STIX]{x1D6EC}\leqslant 0$ , 纽曼猜想互补界 $\unicode[STIX]{x1D6EC}\geqslant 0$ . 在本文中，我们建立了纽曼猜想。论证继续假设矛盾 $\unicode[STIX]{x1D6EC}<0$ 然后分析零点的动态 $H_{t}$ （以 Csordas、Smith 和 Varga 的工作为基础）获得对零点的日益强大的控制 $H_{t}$ 范围中 $\unicode[STIX]{x1D6EC}<t\leqslant 0$ , 直到一个人确定 $H_{0}$ 处于局部平衡状态，从某种意义上说，它们在局部表现（平均而言）就好像它们在算术级数中等距分布，差距保持接近全球平均差距大小。但后一种说法与关于黎曼 zeta 函数的零点局部分布的已知结果不一致，例如蒙哥马利的对相关估计。