论文标题
几乎对有限领域的素数的统一估计
Uniform estimates for almost primes over finite fields
论文作者
论文摘要
我们为有限字段$ \ mathbb {f} _q $建立了一个新的渐近公式，用于$ n $的多项式$ n $的多项式数量。错误项在$ n $和$ q $中均均均匀为$ 0 $，并且$ k $可以超过$ \ log n $。以前，通过Arratia，Barbour和Tavaré的作品，通过Warlimont和Hwang或小$ K $的作品以固定的$ Q $而闻名的渐近公式。 作为应用程序，我们估计$ n $元素随机排列中周期数的总变化距离与$ n $ $ n $的随机质量因子的数量超过$ \ mathbb {f} _q $。距离倾向于$ 0 $以$ 1/（q \ sqrt {\ log n}）$。以前，只有当$ q $固定而$ n $倾向于$ \ infty $，或者$ n $固定并且$ q $倾向于$ \ infty $，根据Arratia，Barbour和Tavaré的结果。
We establish a new asymptotic formula for the number of polynomials of degree $n$ with $k$ prime factors over a finite field $\mathbb{F}_q$. The error term tends to $0$ uniformly in $n$ and in $q$, and $k$ can grow beyond $\log n$. Previously, asymptotic formulas were known either for fixed $q$, through the works of Warlimont and Hwang, or for small $k$, through the work of Arratia, Barbour and Tavaré. As an application, we estimate the total variation distance between the number of cycles in a random permutation on $n$ elements and the number of prime factors of a random polynomial of degree $n$ over $\mathbb{F}_q$. The distance tends to $0$ at rate $1/(q\sqrt{\log n})$. Previously this was only understood when either $q$ is fixed and $n$ tends to $\infty$, or $n$ is fixed and $q$ tends to $\infty$, by results of Arratia, Barbour and Tavaré.
