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Chebyshev'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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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 $p\equiv a(\operatorname{mod}d)$. 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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Additional Information

Pieter Moree
Affiliation: KdV Institute, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands
Email: moree@science.uva.nl

DOI: https://doi.org/10.1090/S0025-5718-03-01536-9
Keywords: Comparative number theory, constants, primes in progression, multiplicative functions
Received by editor(s): November 7, 2001
Received by editor(s) in revised form: May 2, 2002
Published electronically: May 21, 2003
Article copyright: © Copyright 2003 American Mathematical Society

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