关于Mandl不等式的一个注记

New bounds for the sum of the first $n$ prime numbers

In this paper we establish a general asymptotic formula for the sum of the first $n$ prime numbers, which leads to a generalization of the most accurate asymptotic formula given by Massias and Robin. Further we prove a series of results concerning Mandl's inequality on the sum of the first $n$ prime numbers. We use these results to find new explicit estimates for the sum of the first $n$ prime numbers, which improve the currently best known estimates.

关于素计数函数的一个不等式的注记

A remark on an inequality for the prime counting function

我们注意到不等式$0.92 \frac{x}{\log(x)} <π(x)< 1.11 \frac{x}{\log(x)}$并不对所有的$x\ge 30$都成立，与某些文献相反。这些估计$π(x)$最近出现在论文代数数论。

小球与离散不等式

Small Ball and Discrepancy Inequalities

这是Dmitry Bilyk和作者关于ArXiV论文math.CA/0609815的一套综合笔记。本文的重点是一个新的不等式的总和双曲Haar函数在三个变量，扩展了著名的结果J贝克从1987年。这是对小球猜想的改进。在本文中，证明了这一结果，在一个更悠闲的方式和补充说明。此外，背景材料聚集在一起，包括一个完整的证明，必要的调和分析;总结已知的结果对小球不等式; irrational分布;关系与acetrures在逼近论和概率论。