{
  "id": "1112.2398",
  "version": "v1",
  "published": "2011-12-11T20:32:39.000Z",
  "updated": "2011-12-11T20:32:39.000Z",
  "title": "Chebyshev's bias and generalized Riemann hypothesis",
  "authors": [
    "Adel Alamadhi",
    "Michel Planat",
    "Patrick Solé"
  ],
  "comment": "9 pages",
  "journal": "Journal of Algebra, Number Theory: Advances and Applications 8, 1-2 (2013) 41-55",
  "categories": [
    "math.NT"
  ],
  "abstract": "It is well known that $li(x)>\\pi(x)$ (i) up to the (very large) Skewes' number $x_1 \\sim 1.40 \\times 10^{316}$ \\cite{Bays00}. But, according to a Littlewood's theorem, there exist infinitely many $x$ that violate the inequality, due to the specific distribution of non-trivial zeros $\\gamma$ of the Riemann zeta function $\\zeta(s)$, encoded by the equation $li(x)-\\pi(x)\\approx \\frac{\\sqrt{x}}{\\log x}[1+2 \\sum_{\\gamma}\\frac{\\sin (\\gamma \\log x)}{\\gamma}]$ (1). If Riemann hypothesis (RH) holds, (i) may be replaced by the equivalent statement $li[\\psi(x)]>\\pi(x)$ (ii) due to Robin \\cite{Robin84}. A statement similar to (i) was found by Chebyshev that $\\pi(x;4,3)-\\pi(x;4,1)>0$ (iii) holds for any $x<26861$ \\cite{Rubin94} (the notation $\\pi(x;k,l)$ means the number of primes up to $x$ and congruent to $l\\mod k$). The {\\it Chebyshev's bias}(iii) is related to the generalized Riemann hypothesis (GRH) and occurs with a logarithmic density $\\approx 0.9959$ \\cite{Rubin94}. In this paper, we reformulate the Chebyshev's bias for a general modulus $q$ as the inequality $B(x;q,R)-B(x;q,N)>0$ (iv), where $B(x;k,l)=li[\\phi(k)*\\psi(x;k,l)]-\\phi(k)*\\pi(x;k,l)$ is a counting function introduced in Robin's paper \\cite{Robin84} and $R$ resp. $N$) is a quadratic residue modulo $q$ (resp. a non-quadratic residue). We investigate numerically the case $q=4$ and a few prime moduli $p$. Then, we proove that (iv) is equivalent to GRH for the modulus $q$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-12-11T20:32:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "generalized riemann hypothesis",
      "chebyshevs bias",
      "quadratic residue modulo",
      "riemann zeta function",
      "littlewoods theorem"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1112.2398A"
    }
  }
}