Chebyshev’s bias for composite numbers with restricted prime divisors
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- by Pieter Moree;
- Math. Comp. 73 (2004), 425-449
- DOI: https://doi.org/10.1090/S0025-5718-03-01536-9
- Published electronically: May 21, 2003
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Abstract:
Let $\pi (x;d,a)$ denote the number of primes $p\le x$ with $p\equiv a(\operatorname {mod} d)$. Chebyshev’s bias is the phenomenon for which “more often” $\pi (x;d,n)>\pi (x;d,r)$, 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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Bibliographic Information
- Pieter Moree
- Affiliation: KdV Institute, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands
- MR Author ID: 290905
- Email: moree@science.uva.nl
- Received by editor(s): November 7, 2001
- Received by editor(s) in revised form: May 2, 2002
- Published electronically: May 21, 2003
- © Copyright 2003 American Mathematical Society
- 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
- MathSciNet review: 2034131