Chebyshev’s bias for products of k primes

Meng, Xianchang

doi:10.2140/ant.2018.12.305

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Chebyshev's bias for products of $k$ primes

Xianchang Meng

Vol. 12 (2018), No. 2, 305–341

DOI: 10.2140/ant.2018.12.305

Abstract

For any $k \geq 1$ , we study the distribution of the difference between the number of integers $n \leq x$ with $ω (n) = k$ or $Ω (n) = k$ in two different arithmetic progressions, where $ω (n)$ is the number of distinct prime factors of $n$ and $Ω (n)$ 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 $Ω (n) = k$ have preference for quadratic nonresidue 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 $ω (n) = k$ always have preference for quadratic residue classes. Moreover, as $k$ increases, the biases become smaller and smaller for both of the two cases.

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Keywords

Chebyshev's bias, Dirichlet $L$-function, Hankel contour, generalized Riemann hypothesis

Mathematical Subject Classification 2010

Primary: 11M26

Secondary: 11M06, 11N60

Milestones

Received: 6 October 2016

Revised: 26 September 2017

Accepted: 30 October 2017

Published: 13 May 2018

Authors

Xianchang Meng
Centre de Recherches Mathématiques Université de Montréal Montréal, QC Canada	Department of Mathematics University of Illinois at Urbana-Champaign Urbana, IL USA

Vol. 12, No. 2, 2018