AI 中文总结
本文为默滕斯和与乘积提供了新的更优界，运用新技巧改进了相关研究成果，还证明了该默滕斯和的全新精确黎曼-吉南显式公式，结果可用于多领域应用。
AI 中文摘要
本文为默滕斯和与乘积提供了新的界，包括$\boldsymbol{\textstyle \text{sum}_{p \text{≤} x} p^{-1}}$和$\boldsymbol{\textstyle \text{prod}_{p \text{≤} x} (1-\frac{1}{p})}$，在所有范围内为这些和提供了更优的指数型及对数型界。这些带权素数和与乘积曾被Rosser和Schoenfeld（1962）广泛研究，应用于数论、密码学、组合数学等诸多领域。本文提供了大量表格，可用于这类应用。本文的主要新思想是对ζ函数零点的带权和给出了精确界，我们采用了Fiori-Kadiri-Swidinsky（2023）的新技巧，该技巧依赖Kadiri-Lumley-Ng（2018）关于$N(\boldsymbol{\text{σ}},T)$的最新显式零点密度估计。本文中带权零点和的界与技巧有望在其他算术应用中发挥作用。我们的主定理显著改进了Vanlalngaia（2017）的指数衰减结果，修正了Dusart（2018）的研究，填补了文献空白，且结果形式便于未来改进。此外，我们还证明了默滕斯和$\boldsymbol{\textstyle \text{sum}_{p \text{≤} x} p^{-1}}$的精确“黎曼-吉南显式公式”，该公式似乎是全新的。
英文摘要
In this article we provide new bounds for the Mertens sums and products including $\sum_{p \le x} p^{-1}$ and $\prod_{p \le x} (1-\frac{1}{p})$ which provide superior exponential and log type bounds for these sums in all ranges. These weighted prime number sums and products were extensively studied by Rosser and Schoenfeld (1962) and are employed in a wide range of applications in number theory, cryptography, and combinatorics. Extensive tables are provided in this article which will be useful for these types of applications. The main new ideas in this article are sharp bounds for weighted sums of zeros of zeros of the zeta function. We make use of a novel technique of Fiori-Kadiri-Swidinsky (2023) which relies on a recent explicit zero-density estimate for $N(σ,T)$ of Kadiri-Lumley-Ng (2018). The bounds and techniques in this article for weighted zeros sums will likely be useful in many other arithmetic applications. Our main theorem significantly improves the exponential decay result of Vanlalngaia (2017) and fills a gap in the literature by correcting work of Dusart (2018). The results are also presented in a way that are amenable to future improvements. In addition, we prove an exact ``Riemann-Guinand explicit formula" for the Mertens sum $\sum_{p \le x} p^{-1}$ that appears to be new.
Comments37 pages, 19 pages tables
