Vol. 12, No. 2, 2018

Download this article
Download this article For screen
For printing
Recent Issues

Volume 13
Issue 8, 1765–1981
Issue 7, 1509–1763
Issue 6, 1243–1507
Issue 5, 995–1242
Issue 4, 749–993
Issue 3, 531–747
Issue 2, 251–530
Issue 1, 1–249

Volume 12, 10 issues

Volume 11, 10 issues

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the Journal
Editorial Board
Subscriptions
Editors' Interests
Submission Guidelines
Submission Form
Editorial Login
Ethics Statement
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Author Index
To Appear
 
Other MSP Journals
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