We prove an extension of the Landau-Gonek formula. As an application, unconditionally, we recover an important consequence of Montgomery's work on the pair correlation that previously was known under the Riemann Hypothesis. We proceed to show that at least two-thirds of the zeros of the zeta function are simple under a milder assumption than the Riemann Hypothesis. We will also prove a corollary on the value-distribution of Dirichlet polynomials.

中文翻译：

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Landau-Gonek 公式的扩展

我们证明了 Landau-Gonek 公式的扩展。作为一个应用，我们无条件地恢复了蒙哥马利关于先前在黎曼假设下已知的对相关性工作的重要结果。我们继续证明，在比黎曼假设更温和的假设下，zeta 函数的至少三分之二的零点是简单的。我们还将证明狄利克雷多项式的值分布的一个推论。