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This is the Prime Pages'
interface to our BibTeX database. Rather than being an exhaustive database,
it just lists the references we cite on these pages. Please let me know of any errors you notice.References: [ Home  Author index  Key index  Search ]
 Moree2002
 P. Moree, "Chebyshev's bias for composite numbers with restricted prime divisors," (2001) 26 pages.
Abstract:
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 nonresidue 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 LandauRamanujan 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), 193214).
[Available on the web at http://arXiv.org/abs/math/0112100.]
