{
  "id": "2203.12266",
  "version": "v1",
  "published": "2022-03-23T08:29:29.000Z",
  "updated": "2022-03-23T08:29:29.000Z",
  "title": "Chebyshev's Bias against Splitting and Principal Primes in Global Fields",
  "authors": [
    "Miho Aoki",
    "Shin-ya Koyama"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "A reason for the emergence of Chebyshev's bias is investigated. The Deep Riemann Hypothesis (DRH) enables us to reveal that the bias is a natural phenomenon for making a well-balanced disposition of the whole sequence of primes, in the sense that the Euler product converges at the center. By means of a weighted counting function of primes, we succeed in expressing magnitudes of the deflection by a certain asymptotic formula under the assumption of DRH, which gives a new formulation of Chebyshev's bias. For any Galois extension of global fields and for any element $\\sigma$ in the Galois group, we establish a criterion of the bias of primes whose Frobenius elements are equal to $\\sigma$ under the assumption of DRH. As an application we obtain a bias toward non-splitting and non-principle primes in abelian extensions under DRH. In positive characteristic cases, DRH is proved, and all these results hold unconditionally.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-23T08:29:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "chebyshevs bias",
      "global fields",
      "principal primes",
      "deep riemann hypothesis",
      "euler product converges"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}