Chebyshev's bias for products of k primes
Speaker :
Time :
Location :
浙江大学数学科学学院九十周年院庆系列活动之五十六
Title: Chebyshev's bias for products of k primes
Speaker: Prof. Xianchang Meng (McGill University)
Time: 10:00-12:00am, July 17
Location: Room 200-9, Sir Shaw Run Run Business Administration building, School of Mathematical Sciences, Yuquan Campus
Abstract: For any $kgeq 1$, we derive a formula for the difference between the number of integers $nleq x$ with $omega(n)=k$ or $Omega(n)=k$ in two different arithmetic progressions, where $omega(n)$ is the number of distinct prime factors of $n$ and $Omega(n)$ is the number of prime factors of $n$ counted with multiplicity. Under some reasonable assumptions, we show that, if $k$ is odd, the integers with $Omega(n)=k$ have preference for quadratic non-residue classes; and if $k$ is even, such integers have preference for quadratic residue classes. This result confirms a conjecture of Hudson. However, the integers with $omega(n)=k$ always have preference for quadratic residue classes. Moreover, as $k$ increases, the biases become
smaller and smaller for both cases.
Contact Person: Dongwen Liu (maliu@zju.edu.cn)