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bibkey: hedenmalm1997hilbert authors: Hakan Hedenmalm, Peter Lindqvist, and Kristian Seip year: 1997 title: A Hilbert space of Dirichlet series and systems of dilated functions in L2(0,1) doi: 10.1215/S0012-7094-97-08601-4 claim: Square-summable Dirichlet-series Hilbert spaces, their half-plane, zeta kernel, and coefficient translations. strata_touched:

  • D5/S3/Weil/LabeledZeta
  • D5/S3/Weil/SpectralHilbert
  • D5/S3/Weil/SpectralDynamics
  • D5/S3/Zeros/EulerWindows
  • D5/S3/Zeros/SpectralShift license: citation-only triage: anchor

A Hilbert Space of Dirichlet Series

Hakan Hedenmalm, Peter Lindqvist, and Kristian Seip model Dirichlet series f(s) = sum a_n n^(-s) through their square-summable coefficient vectors. That coefficient-space model is the literature anchor for the labeled vector used by D5/S3/Weil/LabeledZeta.labeled_zeta_vector_ne_zero.

The same model anchors the coefficient Hilbert space, the sharp Re s > 1/2 summability boundary, and the zeta kernel formalized in D5/S3/Weil/SpectralHilbert. The repository’s mirror-resonance algebra is an internal consequence of that kernel formula, not a claim attributed to the paper.

It also anchors the coefficient mechanics used by D5/S3/Weil/SpectralDynamics: vertical translation multiplies the nth coefficient by n^(-it), while a positive real translation multiplies it by n^(-delta). Unit modulus in the first case and modulus at most one in the second give the formal norm laws. The paper is not cited for the repository’s unbounded-generator language, zero-resonance terminology, or unified critical-line declaration.

The paper does not use the repository’s ledger vocabulary or state the Lean declaration verbatim. The formal nonvanishing result is the immediate coordinate consequence that the identity coefficient is one.

The same coefficient model is contextual background for the PZG coordinate sum and multiplicative address pullback in the two Zeros modules listed above. The exact PrimeAxisTable encoding and its pointwise backward-shift identity are repository translations, not claims attributed verbatim to the paper.

Search log

  • 2026-07-16: Queried NyxID/Tavily for "A Hilbert space of Dirichlet series and systems of dilated functions" coefficients constant term DOI. The arXiv and Semantic Scholar results identified the coefficient model, Duke Mathematical Journal 86 (1997), pages 1-37, arXiv math/9512211, and DOI 10.1215/S0012-7094-97-08601-4.
  • 2026-07-16: Queried Riemann zeta functional equation conjugation symmetry critical line Re(s)=1/2 scholarly article DOI. Results were encyclopedia, video, and lecture-note pages rather than a DOI-bearing formal source, so the exact mirror fixed-point declaration remains repo-derived.
  • 2026-07-16: Queried the exact phrase "scaling ledger" zeta reflection "1 - conjugate". No scholarly match for the repository’s scaling-ledger formulation was returned; the mirror reversal declaration is repo-derived.
  • 2026-07-16: Five initial proxy attempts in the preceding search pass sent the JSON object with the wrong transport shape and received HTTP 422. Reissuing JSON on stdin with Content-Type: application/json produced the results above; no bibliographic conclusion was drawn from failed requests.
  • 2026-07-17: Queried NyxID/Tavily for Hedenmalm Lindqvist Seip Hilbert space of Dirichlet series reproducing kernel zeta s conjugate w DOI. Results identified the same Duke paper, DOI, and arXiv record; the EMS metadata described the square-summable coefficient space, its analytic domain Re s > 1/2, and the reproducing kernel obtained by evaluating zeta at the summed and conjugated parameters.
  • 2026-07-17: Queried Hedenmalm Lindqvist Seip reproducing kernel zeta s plus conjugate w Hilbert space Dirichlet series formula. A later survey of Hilbert spaces of Dirichlet series displayed the coefficient kernel series and cited the Duke paper with the same DOI. No source used the repository’s mirror-resonance terminology, so that algebra remains repo-derived.
  • 2026-07-17: Queried Hedenmalm Lindqvist Seip Hilbert space Dirichlet series square summable coefficients analytic half plane Re s greater than one half. Independent survey and research sources explicitly attributed the square-summable coefficient definition and the analytic half-plane Re s > 1/2 to Hedenmalm, Lindqvist, and Seip.
  • 2026-07-17: Queried "Hilbert space of Dirichlet series" translation semigroup imaginary translations unitary group. The Hedenmalm paper and a Dirichlet-series Hilbert-space survey were returned; the survey explicitly writes a vertical translate with coefficients a_n exp(-it log n) and records the coefficient norm through vertical translates.
  • 2026-07-17: Queried Dirichlet series Hilbert H2 translation operator f(s+sigma) contraction semigroup coefficient multiplier n power. The results again described the Hedenmalm coefficient space and vertical translates, together with an operator-theory abstract on isometric shift semigroups. They support the coordinate multipliers, but not the source’s stronger unbounded self-adjoint generator and reverse-domain prose.
  • 2026-07-17: Queried Riemann zeta zeros reflection conjugation quartet resonance reproducing kernel and "critical line" reflection fixed line unitarity self resonance l2 boundary half-density Riemann zeta. Results restated the standard reflection and conjugation symmetries, but none used the repository’s kernel-resonance quartet terminology or combined its four critical-line predicates. Those two Describe nodes are therefore repo-derived.
  • 2026-07-17: Six initial NyxID/Tavily calls double-encoded the JSON request body and received HTTP 422. Reissuing raw JSON on stdin with Content-Type: application/json produced the search results above; no bibliographic conclusion was drawn from the failed requests.
  • 2026-07-18: Queried NyxID/Tavily for Hardy space Dirichlet series reproducing kernel eigenvector adjoint multiplication operator prime shifts DOI, Hilbert space Dirichlet series coefficient backward shift eigenvector n to minus s reproducing kernel, and the Hedenmalm-Lindqvist-Seip title with shift and eigenvector terms. Results confirmed the square-summable coefficient model, zeta reproducing kernel, multiplier setting, and the general adjoint-multiplier kernel-eigenvector principle. No result matched the repository’s exact multi-axis PrimeAxisTable pullback or its claimed bundled divisibility operator, so SpectralShift is marked repo-derived and cites this note only as context.

Verified locator

  • DOI: https://doi.org/10.1215/S0012-7094-97-08601-4
  • arXiv: https://arxiv.org/abs/math/9512211