论文标题
奇异系列的总和，带有大型集和素数的尾巴
Sums of singular series with large sets and the tail of the distribution of primes
论文作者
论文摘要
1976年，加拉格尔（Gallagher）表明，在Prime $ k $的艰难木材猜想中，这意味着在日志大小间隔中分布素数是Poissonian。他这样做是通过将奇异系列常数的平均值计算出在$ h \ to \ infty $中包含的固定尺寸$ k $的不同集合的平均值，然后使用此平均值来计算Primes分布的时刻。在本文中，我们研究平均$ k $相对较大的$ h $。然后，我们将这些平均值应用于分布的尾部。例如，我们表明，假设有适当的硬木构想和参数的某些范围，间隔$ [n，n +λ\ log x] $带有$ n \ le x $至少包含$ k $ primes是$ \ ll x \ exp（-k/（λe）。
In 1976, Gallagher showed that the Hardy--Littlewood conjectures on prime $k$-tuples imply that the distribution of primes in log-size intervals is Poissonian. He did so by computing average values of the singular series constants over different sets of a fixed size $k$ contained in an interval $[1,h]$ as $h \to \infty$, and then using this average to compute moments of the distribution of primes. In this paper, we study averages where $k$ is relatively large with respect to $h$. We then apply these averages to the tail of the distribution. For example, we show, assuming appropriate Hardy--Littlewood conjectures and in certain ranges of the parameters, the number of intervals $[n,n +λ\log x]$ with $n\le x$ containing at least $k$ primes is $\ll x\exp(-k/(λe)).$
