{
  "id": "0704.3448",
  "version": "v1",
  "published": "2007-04-26T18:21:22.000Z",
  "updated": "2007-04-26T18:21:22.000Z",
  "title": "Finite Euler products and the Riemann Hypothesis",
  "authors": [
    "S. M. Gonek"
  ],
  "comment": "4 figures",
  "categories": [
    "math.NT",
    "math.CV"
  ],
  "abstract": "We show that if the Riemann Hypothesis is true, then in a region containing most of the right-half of the critical strip, the Riemann zeta-function is well approximated by short truncations of its Euler product. Conversely, if the approximation by products is good in this region, the zeta-function has at most finitely many zeros in it. We then construct a parameterized family of non-analytic functions with this same property. With the possible exception of a finite number of zeros off the critical line, every function in the family satisfies a Riemann Hypothesis. Moreover, when the parameter is not too large, they have about the same number of zeros as the zeta-function, their zeros are all simple, and they \"repel\". The structure of these functions makes the reason for the simplicity and repulsion of their zeros apparent and suggests a mechanism that might be responsible for the corresponding properties of the zeta-function's zeros. Computer evidence suggests that the zeros of functions in the family are remarkably close to those of the zeta-function (even for small values of the parameter), and we show that they indeed converge to them as the parameter increases. Furthermore, between zeros of the zeta-function, the moduli of functions in the family tend to twice the modulus of the zeta-function. Both assertions assume the Riemann Hypothesis. We end by discussing analogues for other L-functions and show how they give insight into the study of the distribution of zeros of linear combinations of L-functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-04-26T18:21:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M26"
    ],
    "keywords": [
      "riemann hypothesis",
      "finite euler products",
      "non-analytic functions",
      "short truncations",
      "finite number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0704.3448G"
    }
  }
}