## Algebra & Number Theory

### Chebyshev's bias for products of $k$ primes

Xianchang Meng

#### 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.

#### Article information

Source
Algebra Number Theory, Volume 12, Number 2 (2018), 305-341.

Dates
Revised: 26 September 2017
Accepted: 30 October 2017
First available in Project Euclid: 23 May 2018

https://projecteuclid.org/euclid.ant/1527040846

Digital Object Identifier
doi:10.2140/ant.2018.12.305

Mathematical Reviews number (MathSciNet)
MR3803705

Zentralblatt MATH identifier
06880890

#### Citation

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

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