Abstract. Let π(x; d, a) denote the number of primes p ≤ x with p ≡ a(mod d). 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. Ifπ(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. 1.

1. INTRODUCTION. There’s nothing quite like a day at the races....The quickening of the pulse as the starter’s pistol sounds, the thrill when your favorite contestant speeds out into the lead (or the distress if another contestant dashes out ahead of yours), and the accompanying fear (or hope) that the leader might change. And what if the race is a marathon? Maybe one of the contestants will be far stronger than the others, taking

We examine the effects of certain hypothetical configurations of zeros of Dirichlet L-functions lying off the critical line on the distribution of primes in arithmetic progressions. 1.

bias for composite numbers with restricted prime divisors

Abstract. It has been well-observed that an inequality of the type π(x; q, a)> π(x; q, b) is more likely to hold if a is a non-square modulo q and b is a square modulo q (the socalled “Chebyshev Bias”). For instance, each of π(x; 8, 3), π(x; 8, 5), and π(x; 8, 7) tends to be somewhat larger than π(x; 8, 1). However, it has come to light that the tendencies of these three π(x; 8, a) to dominate π(x; 8, 1) have different strengths. A related phenomenon is that the six possible inequalities of the form π(x; 8, a1)> π(x; 8, a2)> π(x; 8, a3) with {a1, a2, a3} = {3, 5, 7} are not all equally likely—some orderings are preferred over others. In this paper we discuss these phenomena, focusing on the moduli q = 8 and q = 12, and we explain why the observed asymmetries (as opposed to other possible asymmetries) occur. 1. Background Let π(x; q, a) denote the number of primes not exceeding x that are congruent to a modulo q. We have known since the work of Dirichlet that the two counting functions π(x; q, a) and π(x; q, b) are asymptotically equal as x tends to infinity (as long as a and b are both coprime to q). However, more complicated behavior emerges when we compare these counting functions for finite values of x. Imagine π(x; q, a) and π(x; q, b) as representing the