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Cross-Energy of Dirichlet Partial Sums. Part 16: How many positive directions does the Archimedean place have? A counting law with a boundary phase

A mix of human-written and AI-generated textHuman understanding: all partsmath.NT — Number Theorymath.CV — Complex Variables

Contributed by Pedro Caceres ↗

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

Restrict the archimedean part of Weil's quadratic form, for ζ\zeta or for a primitive real Dirichlet LL-function of conductor qq and parity κ\kappa, to test functions supported in an interval of length aa, and let n+(a)n_+(a) be the number of its positive directions, the directions that the pole and the primes must absorb. A Szeg\H{o}-type argument gives n+(a)=t0a/π+o(a)n_+(a)=t_0a/\pi+o(a), where t0t_0 is the zero of ϑ′\vartheta'. We observe that the form is Λ−ψκ(−Δ)\Lambda-\psi_\kappa(-\Delta) with ψκ\psi_\kappa a complete Bernstein function (proved) and, using the boundary phase of Kaleta, Kwa\'snicki and Ma{\l}ecki, we state, as a conjecture, the asymptotic law \[ n_+(a)=\Big\lfloor\frac{t_0a}\pi+\frac{2\varpi_\kappa(t_0)}\pi\Big\rfloor , \] with an explicit phase ϖκ\varpi_\kappa that depends only on the parity, the conductor entering only through t0t_0. We test it in seven cases, ζ\zeta and χ−3,χ−4,χ5,χ−7,χ8,χ−8\chi_{-3},\chi_{-4},\chi_5,\chi_{-7},\chi_8,\chi_{-8}. On the finest mesh the shifts of the jumps m=2,…,12m=2,\dots,12 bracket the prediction 2ϖκ(t0)/π2\varpi_\kappa(t_0)/\pi, which ranges from 0.180.18 to 0.590.59, to within 11 to 3⋅10−33\cdot10^{-3} for the characters and 3⋅10−23\cdot10^{-2} for ζ\zeta. A second, independent implementation extrapolates in the mesh on three to nine jumps per case; the extrapolated shifts lie above the prediction by 0.20.2 to 2.3⋅10−42.3\cdot10^{-4} in every case, a one-sided offset of the size of the extrapolation error, which we have checked to be a bias in one case only. The first jump is an exception, and for ζ\zeta the shift reaches its limit only after about ten jumps. A smoothed count removes the integer part and leaves ``degree plus constant'' with the boundary phase as constant; on a circle the constant is carried by the end points and cancels between an arc and its complement; and treating −log⁡q-\log q as a degree, q↔1/qq\leftrightarrow1/q acts on the band edge as t0↦4π2/t0t_0\mapsto4\pi^2/t_0 to leading order, with ζ\zeta at the self-dual point t0≈2πt_0\approx2\pi. We record what the law implies for the signature of Weil's form and where it stands with respect to the Riemann--Roch formulas for Spec⁡Z‾\overline{\operatorname{Spec}\mathbb Z}. The law is a conjecture supported by these computations. The Bernstein structure is proved here; the first-order count is quoted from Part~14 and its proof is not repeated.

Author comments: Revised 10 Nov 2026 after external examination (see CHANGES_v2.md). 9 pages, 1 figure; Part 16 of the 47-part book-series Cross-Energy of Dirichlet Partial Sums (Parts 0-46), block III (The margin and the constant B-hat); formerly a standalone note (v1.0). Part 0 is the guide.

Provenance statement

The research programme is the author's. The paper was written in November 2026 as a standalone note (v1.0) and is Part 16 of the book-series Cross-Energy of Dirichlet Partial Sums; this is v1.1 (10 November 2026). It builds on Part 14 of the series, from which the first-order count (Theorem 1) is quoted without re-proof. Claude (Anthropic) was used throughout as a research and writing aid: to derive and check proofs, to write and run the computations, to search the literature, and to draft text and figures. On 10 November 2026 the paper was examined twice by independent AI referee sessions that had not seen it being written; each report listed errors and recomputed the numbers it could, and this version contains the corrections from both rounds. It has not been refereed by a human reviewer. The author checked the results and is responsible for the content. The counting law is a conjecture supported by computations; the main computation was re-implemented from a separate derivation as a check, and the identification of the smooth constant with Widom's second-order term is numerical (script widom_check.py, written by the referee session and rerun by the author).

References

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Version history

  1. v1 · 9 pagesSubmitted by Pedro CaceresRevised 10 Nov 2026 after external examination (see CHANGES_v2.md). 9 pages, 1 figure; Part 16 of the 47-part book-series Cross-Energy of Dirichlet Partial Sums (Parts 0-46), block III (The margin and the constant B-hat); formerly a standalone note (v1.0). Part 0 is the guide.CurrentNov 11, 2026