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Chebyshev’s bias for products of k primes and Greg Martin’s conjecture

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Xianchang Meng
Universität Göttingen/MPIM
Wed, 2019-08-28 14:30 - 15:30
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

For any $k\geq 1$, we study the distribution of the difference between the number of integers $n\leq x$ 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. As an application, we give strong evidence for a conjecture of Greg Martin, which concerns the total number of prime factors in different arithmetic progressions.

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