{
  "id": "0912.4908",
  "version": "v3",
  "published": "2009-12-24T20:18:31.000Z",
  "updated": "2011-07-21T22:38:53.000Z",
  "title": "Inequities in the Shanks-Renyi Prime Number Race: An asymptotic formula for the densities",
  "authors": [
    "Daniel Fiorilli",
    "Greg Martin"
  ],
  "comment": "76 pages. Corrected a numerical error in Proposition 2.14 and its consequences in Section 5. Paper has been accepted to Crelle",
  "categories": [
    "math.NT"
  ],
  "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 $\\pi(x;q,a)$ to exceed $\\pi(x;q,b)$. Rubinstein and Sarnak defined $\\delta(q;a,b)$ to be the logarithmic density of the set of positive real numbers $x$ for which this inequality holds; intuitively, $\\delta(q;a,b)$ is the \"probability\" that $\\pi(x;q,a) > \\pi(x;q,b)$ when $x$ is \"chosen randomly\". In this paper, we establish an asymptotic series for $\\delta(q;a,b)$ 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 $\\delta(q;a,b)$ converges to $\\frac12$ as $q$ grows, these evaluations allow us to compare the various density values $\\delta(q;a,b)$ 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 $\\delta(q;a,1) < \\delta(q;a',1)$ for all but finitely many moduli $q$ for which both $a$ and $a'$ are nonsquares. Finally, we establish rigorous numerical bounds for these densities $\\delta(q;a,b)$ and report on extensive calculations of them.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2011-07-21T22:38:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N13",
      "11Y35"
    ],
    "keywords": [
      "shanks-renyi prime number race",
      "asymptotic formula",
      "inequities",
      "residue classes",
      "error term smaller"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 76,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0912.4908F"
    }
  }
}