标题： 无条件的蒙哥马利定理关于黎曼zeta函数零点的对相关性 显示英文标题

标题： An unconditional Montgomery Theorem for Pair Correlation of Zeros of the Riemann Zeta Function

摘要： 假设黎曼猜想（RH），蒙哥马利证明了一个关于黎曼zeta函数零点对相关性的定理。 这个定理的一个结果是，假设RH成立，至少$67.9\%$个非平凡零点是简单的。 在这里，我们得到了蒙哥马利定理的无条件形式，并展示了如何应用它来证明关于简单零点的结果：假设所有满足$\rho=\beta+i\gamma$的黎曼zeta函数的零点，其中$T^{3/8}<\gamma\le T$满足$|\beta-1/2|<1/(2\log T)$，%位于细长的盒子$\{s=\sigma +it: |\sigma-1/2|<1/(2\log T),\ T^{3/8}<t\le T\}$中，那么，当$T$趋向于无穷大时，至少$61.7\%$个这些零点是简单的。 证明的方法既不要求也不提供任何关于这些零点是否在临界线$\beta=1/2$上的任何信息。 我们在更强的零点密度假设下也得到了相同的结果。 显示英文摘要

摘要： Assuming the Riemann Hypothesis (RH), Montgomery proved a theorem concerning pair correlation of zeros of the Riemann zeta-function. One consequence of this theorem is that, assuming RH, at least $67.9\%$ of the nontrivial zeros are simple. Here we obtain an unconditional form of Montgomery's theorem and show how to apply it to prove the following result on simple zeros: Assuming all the zeros $\rho=\beta+i\gamma$ of the Riemann zeta-function such that $T^{3/8}<\gamma\le T$ satisfy $|\beta-1/2|<1/(2\log T)$, %lie in the thin box $\{s=\sigma +it: |\sigma-1/2|<1/(2\log T),\ T^{3/8}<t\le T\}$, then, as $T$ tends to infinity, at least $61.7\%$ of these zeros are simple. The method of proof neither requires nor provides any information on whether any of these zeros are on or not on the critical line where $\beta=1/2$. We also obtain the same result under the weaker assumption of a strong zero-density hypothesis.