ABSTRACT. Chebyshev was the first to observe a bias in the distribution of primes in residue classes. The general phenomenon is that if a is a nonsquare (mod q) and b is a square (mod q), then there tend to be more primes congruent to a (mod q) than b (mod q) in initial intervals of the positive integers; more succinctly, there is a tendency for π(x; q, a) to exceed π(x; q, b). Rubinstein and Sarnak defined δ(q; a, b) to be the logarithmic density of the set of positive real numbers x for which this inequality holds; intuitively, δ(q; a, b) is the "probability" that π(x; q, a) > π(x; q, b) when x is "chosen randomly". In this paper, we establish an asymptotic series for δ(q; a, b) that can be instantiated with an error term smaller than any negative power of q. This asymptotic formula is written in terms of a variance V (q; a, b) that is originally defined as an infinite sum over all nontrivial zeros of Dirichlet L-functions corresponding to characters (mod q); we show how V (q; a, b) can be evaluated exactly as a finite expression. In addition to providing the exact rate at which δ(q; a, b) converges to 1 2 as q grows, these evaluations allow us to compare the various density values δ(q; a, b) as a and b vary modulo q; by analyzing the resulting formulas, we can explain and predict which of these densities will be larger or smaller, based on arithmetic properties of the residue classes a and b (mod q). For example, we show that if a is a prime power and a is not, then δ(q; a, 1) < δ(q; a , 1) for all but finitely many moduli q for which both a and a are nonsquares. Finally, we establish rigorous numerical bounds for these densities δ(q; a, b) and report on extensive calculations of them, including for example the determination of all 117 density values that exceed 9 10 .

We derive explicit upper bounds for various counting functions for primes in arithmetic progressions. By way of example, if q and a are integers with gcd(a, q) = 1 and 3 ≤ q ≤ 10 5 , and θ(x; q, a) denotes the sum of the logarithms of the primes p ≡ a (mod q) with p ≤ x, we show that θ(x; q, a) − x/ϕ(q) < 1 160x log x for all x ≥ 8 • 10 9 , with significantly sharper constants obtained for individual moduli q. We establish inequalities of the same shape for the other standard primecounting functions π(x; q, a) and ψ(x; q, a), as well as inequalities for the nth prime congruent to a (mod q) when q ≤ 1200. For moduli q > 10 5 , we find even stronger explicit inequalities, but only for much larger values of x. Along the way, we also derive an improved explicit lower bound for L(1, χ) for quadratic characters χ, and an improved explicit upper bound for exceptional zeros.

Rubinstein and Sarnak investigated systems of inequalities of the form π(x; q, a 1 ) > · · · > π(x; q, a r ), where π(x; q, a) denotes the number of primes up to x that are congruent to a mod q. They showed, under standard hypotheses on the zeros of Dirichlet L-functions mod q, that the set of positive real numbers x for which these inequalities hold has positive (logarithmic) density δ q;a 1 ,...,ar > 0. They also discovered the surprising fact that a certain distribution associated with these densities is not symmetric under permutations of the residue classes a i in general, even if the a i are all squares or all nonsquares mod q (a condition necessary to avoid obvious biases of the type first observed by Chebyshev). This asymmetry suggests, contrary to prior expectations, that the densities δ q;a 1 ,...,ar themselves vary under permutations of the a i .In this paper, we derive (under the hypotheses used by Rubinstein and Sarnak) a general formula for the densities δ q;a 1 ,...,ar , and we use this formula to calculate many of these densities when q ≤ 12 and r ≤ 4. For the special moduli q = 8 and q = 12, and for {a 1 , a 2 , a 3 } a permutation of the nonsquares {3, 5, 7} mod 8 and {5, 7, 11} mod 12, respectively, we rigorously bound the error in our calculations, thus verifying that these densities are indeed asymmetric under permutation of the a i . We also determine several situations in which the densities δ q;a 1 ,...,ar remain unchanged under certain permutations of the a i , and some situations in which they are provably different.

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 the lead and running at the head of the pack for the whole race. Or perhaps the race will be more dramatic, with the lead changing again and again for as long as one cares to watch.

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