Sphere-packings, lattices, and groups
Sphere-packings, lattices, and groups
Primes in arithmetic progressions
Mathematics of Computation
The hexagonal versus the square lattice
Mathematics of Computation
The hexagonal versus the square lattice
Mathematics of Computation
Finite Fields and Their Applications
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Let π(x;d,a) denote the number of primes p ≤ x with p ≡ a(modd). Chebyshev's bias is the phenomenon for which "more often" π(x; d, n) π(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 π(x; d, n) ≥ π(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 for a much larger class of constants.