Algebra & Number Theory
- Algebra Number Theory
- Volume 12, Number 2 (2018), 305-341.
Chebyshev's bias for products of $k$ primes
For any , we study the distribution of the difference between the number of integers with or in two different arithmetic progressions, where is the number of distinct prime factors of and is the number of prime factors of counted with multiplicity. Under some reasonable assumptions, we show that, if is odd, the integers with have preference for quadratic nonresidue classes; and if is even, such integers have preference for quadratic residue classes. This result confirms a conjecture of Richard Hudson. However, the integers with always have preference for quadratic residue classes. Moreover, as increases, the biases become smaller and smaller for both of the two cases.
Algebra Number Theory, Volume 12, Number 2 (2018), 305-341.
Received: 6 October 2016
Revised: 26 September 2017
Accepted: 30 October 2017
First available in Project Euclid: 23 May 2018
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 11M26: Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses
Secondary: 11M06: $\zeta (s)$ and $L(s, \chi)$ 11N60: Distribution functions associated with additive and positive multiplicative functions
Meng, Xianchang. Chebyshev's bias for products of $k$ primes. Algebra Number Theory 12 (2018), no. 2, 305--341. doi:10.2140/ant.2018.12.305. https://projecteuclid.org/euclid.ant/1527040846