{
  "id": "2307.05958",
  "version": "v1",
  "published": "2023-07-12T07:02:52.000Z",
  "updated": "2023-07-12T07:02:52.000Z",
  "title": "Chebyshev's bias for Fermat curves of prime degree",
  "authors": [
    "Yoshiaki Okumura"
  ],
  "comment": "14 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this article, we prove that an asymptotic formula on the prime number race for Fermat curves of prime degree is equivalent to a part of the Deep Riemann Hypothesis (DRH), which is a conjecture on the convergence of the partial Euler products of $L$-functions on the critical line. We also show that such equivalences hold for some quotients of Fermat curves. As an application, we compute the orders of zeros at $s=1$ for the second moment $L$-functions of these curves under DRH.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-12T07:02:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "11N05",
      "11G30"
    ],
    "keywords": [
      "fermat curves",
      "prime degree",
      "chebyshevs bias",
      "partial euler products",
      "prime number race"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}