论文标题
Riemann假设的Riesz型标准
Riesz-type criteria for the Riemann hypothesis
论文作者
论文摘要
在1916年，里兹（Riesz）证明了riemann假设等于绑定的$ \ sum_ {n = 1}^\ infty \ infty \ frac {μ（n）} {n^2} \ exp \ exp \ left（ - +ε} \ right）$，as $ x \ rightarrow \ infty $，对于任何$ε> 0 $。大约在同一时间，Hardy和Littlewood在纠正Ramanujan的身份时，为Riemann假设提供了另一个同等标准。在本文中，我们建立了对Hardy和Littlewood的身份的单变量概括，并作为应用程序，为Riemann假设提供了Riesz型标准。特别是，我们获得了Riesz以及Hardy和Littlewood的界限。
In 1916, Riesz proved that the Riemann hypothesis is equivalent to the bound $\sum_{n=1}^\infty \frac{μ(n)}{n^2} \exp\left( - \frac{x}{n^2} \right) = O_ε \left( x^{-\frac{3}{4} + ε} \right)$, as $x \rightarrow\infty$, for any $ε>0$. Around the same time, Hardy and Littlewood gave another equivalent criteria for the Riemann hypothesis while correcting an identity of Ramanujan. In the present paper, we establish a one-variable generalization of the identity of Hardy and Littlewood and as an application, we provide Riesz-type criteria for the Riemann hypothesis. In particular, we obtain the bound given by Riesz as well as the bound of Hardy and Littlewood.
