|Date: Monday, February 19, 2018
Location: 4088 East Hall (4:10 PM to 5:30 PM)
Title: Chebyshev's bias for products of k primes and related applications
Abstract: For any k >= 1, we study the distribution of the difference between the number of integers n <= x with $\omega(n)=k$ or $\Omega(n)=k$ among 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 a conjecture of Greg Martin about the total number of prime factors for integers up to x among different arithmetic progressions.
If time permits, I may mention the generalization of Chebyshev's bias problem to products of irreducible elements in Function Fields. There will be more interesting phenomenon for the function field version due to the existence of real zeros of corresponding L-functions.
Speaker: Xianchang Meng
Institution: McGill University
Event Organizer: Jeff Lagarias