@unpublished{Moree2002,
author={P. Moree},
title={Chebyshev's bias for composite numbers with restricted prime divisors},
abstract={Let $P(x,d,a)$ denote the number of primes $p \le x$ with $p \equiv a (mod
d)$. Chebyshev's bias is the phenomenon that `more often' $P(x;d,n)>P(x;d,r)$
than the other way around, where n is a quadratic non-residue mod $d$ and
$r$ is a quadratic residue mod $d$. If $P(x;d,n) \ge P(x;d,r)$ 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 $p \equiv a (mod d)$.
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
true for a much larger class of constants. In a sequel to this paper the
methods developed here will be used and somewhat refined to resolve a conjecture
from P. Schmutz Schaller to the extent that the hexagonal lattice is `better'
than the square lattice (see p. 201 of Bull. Amer. Math. Soc. 35 (1998),
193-214).},
note={26 pages},
year= 2001,
annote={Available on the web at \url{http://arXiv.org/abs/math/0112100}.}
}
