{
  "id": "2210.03616",
  "version": "v1",
  "published": "2022-10-07T15:17:20.000Z",
  "updated": "2022-10-07T15:17:20.000Z",
  "title": "Evaluation of the multiple zeta values $ζ(2,\\ldots,2,4,2,\\ldots,2)$ via double zeta values, with applications to period polynomial relations and to multiple $t$ values",
  "authors": [
    "Steven Charlton",
    "Adam Keilthy"
  ],
  "comment": "47 pages. Computer readable versions of the full evaluations in Mathematica and pari/gp syntax is included in the arXiv submission",
  "categories": [
    "math.NT"
  ],
  "abstract": "In studying the depth filtration on multiple zeta values, difficulties quickly arise due to a disparity between it and the coradical filtration. In particular, there are additional relations in the depth graded algebra coming from period polynomials of cusp forms for $SL_2(\\mathbb{Z})$. In contrast, a simple combinatorial filtration, the block filtration is known to agree with the coradical filtration, and so there is no similar defect in the associated graded. However, via an explicit evaluation of $\\zeta(2,\\ldots,2,4,2,\\ldots,2)$ as a polynomial in double zeta values, we derive these period polynomial relations as a consequence of an intrinsic symmetry of block graded multiple zeta values in block degree 2. In deriving this evaluation, we find a Galois descent of certain alternating double zeta values to classical double zeta values, which we then apply to give an evaluation of the multiple $t$ values $t(2\\ell,2k)$ in terms of classical double zeta values.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-07T15:17:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M32",
      "11M41",
      "11G99"
    ],
    "keywords": [
      "period polynomial relations",
      "evaluation",
      "classical double zeta values",
      "block graded multiple zeta values",
      "applications"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 47,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}