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- P. Moree, "Chebyshev's bias for composite numbers with restricted prime divisors," (2001) 26 pages.
Let P(x,d,a) denote the number of primes p ≤ x with p ≡ 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) ≥ P(x;d,r) for every x up to some large number, then one expects that N(x;d,n) ≥ N(x;d,r) for every x. Here N(x;d,a) denotes the number of integers n ≤ x such that every prime divisor p of n satisfies p ≡ 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) ≥ 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).
[Available on the web at http://arXiv.org/abs/math/0112100.]