Chebyshev’s bias for products of k primes and total number of prime factors in arithmetic progressions

Speaker: 孟宪昌 (Centre de RecherchesMathematiques, University of Montreal and McGill University)

Venue: B1044

Time: 2017年12月26日   10:00-11:00

Title: Chebyshev’s bias for products of k primes and total number of prime factors in arithmetic progressions.

Abstract: For any $kgeq 1$, we study the distribution of 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 Richard Hudson. However, the integers with $omega(n)=k$ always have preference for quadratic residue classes. As an application of the method developed for settling the above problem, we consider the total number of prime factors for integers up to x among different arithmetic progressions.