Vol. 12, No. 2, 2018

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Chebyshev's bias for products of $k$ primes

Xianchang Meng

Vol. 12 (2018), No. 2, 305–341
Abstract

For any k 1, we study the distribution of the difference between the number of integers n 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.

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