{
  "abstract": "Let $0<γ_1\\leq γ_2\\leq \\cdots$ be the ordinates of the non-trivial zeros of the Riemann zeta function in the upper half-plane, counted with multiplicity, and fix $c>0$. We say that a non-trivial zero $ρ_n=β_n+iγ_n$ is $c$-isolated if $\\min(γ_{n+1}-γ_n, γ_n-γ_{n-1})>2πc/\\log γ_n$. Note that any isolated zero is necessarily simple and on the critical line. We prove unconditionally that there exists a positive constant $c>0$ such that at least $13/20$ of the zeta zeros are $c$-isolated. Moreover, we can show that this proportion is more than $1/2$ when $c=1/6$. Previously, it was not known unconditionally that there are infinitely many isolated zeros. We also show that there are positive proportions of $k$ consecutive ``moderate'' gaps (gaps larger than a constant times the average local gap) between the ordinates of the zeta zeros, for $k=3, 4, 5$. As an application we combine our work on isolated zeros (which we extend to Dirichlet $L$-functions) with an unconditional refinement of the Bombieri--Hejhal method to show unconditionally that at least $31\\%$ of the zeros of a linear combination of two distinct primitive Dirichlet $L$-functions with the same conductor and parity lie on the critical line and have odd multiplicity. The previous unconditional record for this proportion was $7.38\\times 10^{-9}$, which was obtained by Dousselin who refined and made explicit a result of Selberg.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.26841",
  "author_total": 1,
  "authors": [
    {
      "name": "Youness Lamzouri"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fy6bs53ambv42.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.26841",
  "fulltext_url": null,
  "id": "fy6bs53ambv42",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.26841",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7223044069",
  "pdf_url": "https://arxiv.org/pdf/2610.26841",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fy6bs53ambv42.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:43Z",
      "native_id": "2610.26841",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.26841"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Isolated zeros of the Riemann zeta function and zeros of linear combinations of Dirichlet $L$-functions",
  "title_text": "Isolated zeros of the Riemann zeta function and zeros of linear combinations of Dirichlet L-functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:43Z",
  "url": "https://research.draconitic.com/paper/fy6bs53ambv42",
  "venue": "arXiv",
  "versions": []
}