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Chebyshev's bias for products of k primes

编辑:xyx 时间:2018年07月13日 访问次数:240

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)