{
  "abstract": "Starting from the Riemann-Siegel decomposition of $ζ(s)$ given in Siegel's 1932 paper, we introduce a change of variable, $t=I(T)$, that replaces Siegel's coupled pair \"imaginary part $t$ / summation index $m$\" with one real index $T$. We then split Siegel's single remainder integral $R$ into two exact pieces, $R_{1ps}$ and $R_{2ps}$, and prove that $R=R_{1ps}+R_{2ps}$, so that $ζ=Σ_1+R_{1ps}+Σ_2+R_{2ps}$. Our central observation is that each of these remainders is nothing more than one additional, fractional partial summand appended to its Dirichlet sum: $ζ(s)=\\sum_{n=1}^{m}n^{-s}+\\hat{d}_1(m+1)^{-s}+χ(s)\\sum_{n=1}^{m}n^{s-1}+\\hat{d}_2χ(s)(m+1)^{s-1}$, with $\\hat{d}_1,\\hat{d}_2$ real numbers (always positive on the critical line), the fractions of those two summands that are used. As a corollary, when $σ=\\frac{1}{2}$ one has $d_1=d_2$ (equivalently $\\hat{d}_1=\\hat{d}_2$); this fact is formally verified in Lean. Also, with this rescaling the remainder terms are nearly periodic in $T$ with period one, converging to a fixed waveform in the fractional part of $T$. This decomposition of Siegel's $R$ was discovered through experimental mathematics using a spiral visualization of the partial sums, described later in the paper. We also discuss a number of other observations, including what we call the yin yang curves, the zero counting function, and ovals of equal length leg loci.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03728",
  "author_total": 1,
  "authors": [
    {
      "name": "Paul Stahura"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/aw9v84a9ph8nj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03728",
  "fulltext_url": null,
  "id": "aw9v84a9ph8nj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03728",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7220664364",
  "pdf_url": "https://arxiv.org/pdf/2610.03728",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/aw9v84a9ph8nj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.03728",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03728"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The Riemann-Siegel remainder as a fractional summand",
  "title_text": "The Riemann-Siegel remainder as a fractional summand",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/aw9v84a9ph8nj",
  "venue": "arXiv",
  "versions": []
}