ISSN 1088-6842(online) ISSN 0025-5718(print)

Chebyshevâs bias for composite numbers with restricted prime divisors

Author: Pieter Moree
Journal: Math. Comp. 73 (2004), 425-449
MSC (2000): Primary 11N37, 11Y60; Secondary 11N13
DOI: https://doi.org/10.1090/S0025-5718-03-01536-9
Published electronically: May 21, 2003
MathSciNet review: 2034131
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Let $\pi (x;d,a)$ denote the number of primes $p\le x$ with $p\equiv a(\operatorname {mod} d)$. Chebyshevâs bias is the phenomenon for which âmore oftenâ $\pi (x;d,n)>\pi (x;d,r)$, than the other way around, where $n$ is a quadratic nonresidue mod $d$ and $r$ is a quadratic residue mod $d$. If $\pi (x;d,n)\ge \pi (x;d,r)$ for every $x$ up to some large number, then one expects that $N(x;d,n)\ge N(x;d,r)$ for every $x$. Here $N(x;d,a)$ denotes the number of integers $n\le x$ such that every prime divisor $p$ of $n$ satisfies $p\equiv a(\mathrm {mod} d)$. In this paper we develop some tools to deal with this type of problem and apply them to show that, for example, $N(x;4,3)\ge N(x;4,1)$ for every $x$.

In the process we express the so-called second order Landau-Ramanujan constant as an infinite series and show that the same type of formula holds for a much larger class of constants.

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