{
  "abstract": "Let &theta; be the Riemann-Siegel theta function and Z(t) = e i&theta;(t) &zeta;(1 2 + it) Hardy's function. For α &isin; [0, 1) let L α = {t > t 0 : &theta;(t) &equiv; απ (mod π)}, numbered by &theta;(t n) = (n + α)π; L 0 is the sequence of Gram points. We prove that for every α T <tn≤2T &zeta;(1 2 + it n) 2&theta; ′ (t n) = 1 + e −2πiα T 2π + O T 1/4 log T , so that the &theta; ′-weighted mean of (−1) n Z(t n) over L α is 2 cos πα. At α = 0 this is Titch-marsh's theorem on the mean of Hardy's function at the Gram points, for the alternating sum (−1) n Z(g n) = &zeta;(1 2 + ig n), with an error that, combined with Bourgain's sub-convexity bound, is O(N 13/84+&epsilon;); the parity-separated sums over even and over odd Gram points, for which Cao, Tanigawa and Zhai have O(N 1/4 log 3/4 N log log N), are a finer statistic that the present method does not reach. At α = 1 2 , where &zeta;(1 2 + it n) is purely imaginary, the mean is 0. The proof is a one-page contour integration of &zeta;(s)/(1 + λ&chi;(s)), λ = −e 2πiα , whose poles are exactly the points of L α on the critical line; the vertical edges of the contour lie in the half-planes of absolute convergence and contribute T and T /λ with error O(1). The theorem extends to a class of L-functions under explicit analytic hypotheses on the functional equation. We also report how Gram's law depends on the phase: on the first 600 zeros the proportion of lattice intervals containing exactly one zero falls from 0.936 at α = 0 to 0.484 at α = 1 2 , in the direction the mean value suggests.",
  "abstract_shown": true,
  "affiliations": [
    "Universidad de Buenos Aires"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Veltri Pablo",
      "orcid": "0009-0009-6136-6202"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0h8xjhxe7m0hy.bib",
  "cited_by": 0,
  "doi": "10.13140/rg.2.2.24583.36005",
  "fulltext_url": null,
  "id": "0h8xjhxe7m0hy",
  "issns": [],
  "landing_url": "https://zenodo.org/records/23769727",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7224451935",
  "pdf_url": "https://zenodo.org/api/records/23769727/files/gram-lattices-final.pdf/content",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0h8xjhxe7m0hy.ris",
  "sources": [
    {
      "first_seen": "2026-11-12T12:20:57Z",
      "native_id": "23769727",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23769727"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2602|Analytic Number Theory Research",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Mean values of Hardy's function on the Gram lattices of arbitrary phase",
  "title_text": "Mean values of Hardy's function on the Gram lattices of arbitrary phase",
  "type": "preprint",
  "updated_at": "2026-11-12T12:20:57Z",
  "url": "https://research.draconitic.com/paper/0h8xjhxe7m0hy",
  "venue": "Zenodo",
  "versions": []
}