bibkey: coffey2007theta authors: Mark W. Coffey year: 2007 title: Theta and Riemann xi function representations from harmonic oscillator eigensolutions doi: 10.1016/j.physleta.2006.10.055 claim: The completed classical zeta, its xi function, and theta-Mellin representations of the functional equation. strata_touched:

• D5/S3/Zeros/CompletedZeta license: citation-only triage: anchor

Theta and Riemann Xi Function Representations

Mark W. Coffey records the completed classical zeta and develops theta-based integral representations that extend its functional equation. This supplies a literature anchor for the classical completed-zeta and xi facts used in D5/S3/Zeros/CompletedZeta: xi is entire and is invariant under replacing s by 1 - s.

The paper does not state the repository declarations verbatim. The Lean definition is mathlib’s total pole-removed implementation, and the checked proofs invoke mathlib’s completed-zeta theorems rather than formalizing the paper’s harmonic-oscillator, Poisson-summation, or Mellin-transform derivation. The generic zero-orbit and scaling-ledger theorem is repo-derived and is not attributed to this source.

Search log

• 2026-07-18: Queried NyxID/Tavily for Riemann zeta completed xi functional equation analytic continuation scholarly source DOI. Results located the classical completed-zeta formula and the theta/Mellin route, but the broad search mixed reference pages with primary and expository sources.
• 2026-07-18: Queried M W Coffey Theta and Riemann xi function representations harmonic oscillator DOI and then the exact title with doi. The IAEA INIS metadata and ScienceDirect record agreed on the title, journal, PII, and DOI 10.1016/j.physleta.2006.10.055; the abstract states that the paper extends the zeta functional equation and develops completed xi representations.
• 2026-07-18: Queried the 1859 Riemann title and Titchmarsh’s second-edition title for a DOI-bearing locator. Those searches verified the historical and standard-book context but did not return a DOI for either item, so the directly verified Coffey article is used as the machine-addressable anchor.

Verified locator

• DOI: https://doi.org/10.1016/j.physleta.2006.10.055
• arXiv: https://arxiv.org/abs/math-ph/0612086