{
  "abstract": "Let $E_n$ denote the number of alternating permutations of $\\{1,\\dots,n\\}$, equivalently characterized by $\\sum_{n\\ge0}E_nz^n/n!=\\sec z+\\tan z$. For every $q\\ge1$, the sequence $(E_n\\bmod q)_{n\\ge0}$ is eventually periodic; let $d(q)$ and $s(q)$ denote its minimal eventual period and preperiod. For every odd prime $p$, Knuth and Buckholtz proved $d(p)=\\operatorname{lcm}(p-1,4)$ together with \\[ d(p^r)\\mid p^{r-1}d(p), \\qquad s(p^r)\\le r, \\] and Ramassamy conjectured that both bounds are attained for every $r\\ge1$. In this paper, we introduce an algebraic frequency expansion for the Euler zig-zag numbers using Hurwitz series over the coefficient ring $S_r=(\\mathbb Z/p^r\\mathbb Z)[x]/(x^2+1)$. More precisely, the corresponding Hurwitz series is represented as a finite combination of formal exponential modes, in a manner reminiscent of Fourier analysis. Using this expansion, we prove \\[ d(p^r)=p^{r-1}d(p) \\qquad \\text{for every odd prime $p$ and every $r\\ge1$}, \\] thereby establishing Ramassamy's period conjecture. We also disprove the preperiod conjecture by proving \\[ s(5^5)=4<5. \\] Finally, we prove that $5^5$ is the smallest odd prime power for which $s(p^r)\\ne r$, and based on our findings we further conjecture \\[ s(p^r)\\ge r-2 \\] for every odd prime $p$ and every $r\\ge2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.27058",
  "author_total": 1,
  "authors": [
    {
      "name": "Berke Güleç"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/305qymhvsdpxr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.27058",
  "fulltext_url": null,
  "id": "305qymhvsdpxr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.27058",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7204259643",
  "pdf_url": "https://arxiv.org/pdf/2608.27058",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/305qymhvsdpxr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2608.27058",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.27058"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Modular periodicity of the Euler up/down numbers at odd prime powers",
  "title_text": "Modular periodicity of the Euler up/down numbers at odd prime powers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/305qymhvsdpxr",
  "venue": "arXiv",
  "versions": []
}