Name
Abstract | ||
|---|---|---|
Rubinstein and Sarnak investigated systems of inequalities of the form pi (x; q, a(1)) >... > pi (x; q, a(r)), where pi (x; q, b) denotes the number of primes up to x that are congruent to b mod q. They showed, under standard hypotheses on the zeros of Dirichlet F-functions mod q, that the set of positive real numbers x for which these inequalities hold has positive (logarithmic) density delta (q;a1,...,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, in general, even if the ai 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 delta (q;a1,...,ar) themselves vary under permutations of the a(j). Here we derive (under the hypotheses used by Rubinstein and Sarnak) a general formula for the densities delta (q;a1,...,ar), and we use this formula to calculate many of these densities when q less than or equal to 12 and r less than or equal to 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, Il)modl2, respectively, we rigorously bound the error in our calculations, thus verifying that these densities are indeed asymmetric under permutation of the aj. We also determine several situations in which the densities delta (q;a1,...,ar) remain unchanged under certain permutations of the aj, and some situations in which they are provably different. |
Year | DOI | Venue |
|---|---|---|
2000 | 10.1080/10586458.2000.10504659 | EXPERIMENTAL MATHEMATICS |
Keywords | Field | DocType |
Chebyshev's bias,comparative prime number theory,primes in arithmetic progressions,Shanks-Renyi race | Topology,Discrete mathematics,Prime number,Primes in arithmetic progression,Positive real numbers,Mathematical analysis,Permutation,Dirichlet distribution,Logarithm,Congruence (geometry),Mathematics,Chebyshev's bias | Journal |
Volume | Issue | ISSN |
9.0 | 4.0 | 1058-6458 |
Citations | PageRank | References |
1 | 0.45 | 0 |
Authors | ||
2 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Andrey Feuerverger | 1 | 1 | 1.12 |
| Greg Martin | 2 | 1 | 0.45 |