bibkey: bringmann2024periodic
authors: "Kathrin Bringmann, Guoniu Han, Bernhard Heim and Ben Kane"
year: 2024
title: "Periodic sign changes for weakly holomorphic eta-quotients"
doi: 10.48550/arXiv.2409.07164
url: https://arxiv.org/abs/2409.07164v1
claim: "Appendix A Table 1 asks for period-nine signs of normalized Euler quotients, including exponents 9 at 1 and -6 or -10 at 3."
strata_touched:
  - D5/S3/Arith/ModularForms/CubicThetaLattice
  - D5/S3/Arith/ModularForms/CubicThetaCoreProduct
  - D5/S3/Arith/ModularForms/CubicThetaIdentities
  - D5/S3/Arith/ModularForms/EtaSignK6
  - D5/S3/Combinatorics/Partitions/RestrictedEulerCancellation
  - D5/S3/Combinatorics/Partitions/EulerNineComparison
  - D5/S3/Combinatorics/PowerSeries/LaurentSpecialization
  - D5/S3/Arith/ModularForms/WeightedJacobiSpecialization
  - D5/S3/Arith/ModularForms/EulerPentagonalTrisection
  - D5/S3/Arith/ModularForms/EulerNineDissection
  - D5/S3/Arith/ModularForms/EulerNineSquareDissection
  - D5/S3/Arith/ModularForms/EtaSignK10
license: citation-only
triage: anchor

The exact k6 normalized Euler sign question

Verified locator

DOI: 10.48550/arXiv.2409.07164

URL: https://arxiv.org/abs/2409.07164v1

The inspected v1 primary TeX is single-source.tex, extracted text SHA256 cf5c92e1e04a9310d3ef71f6bd839274bd2c611240fb26d4ae988098a42717ab. The introduction defines the coefficients by normalized Euler products. Appendix A Table 1, HTML locator A1.T1.4, contains the exact entry 1^9 3^(-6) with the pattern +-+--++-+, starting at residue zero. The author-hosted manuscript dated May 21, 2025, printed p. 21, repeats this entry; its PDF SHA256 is 8e07e6cb135e2bbb3c9caa082ed0262c2aa9b6b1bfccd6f0162cc570191ebace. The publisher’s typeset table was not inspected.

Exact scope and formal anchors

Write E(X)=prod_(j>=1)(1-X^j). The question concerns every natural index of F6=E(X)^9/E(X^3)^6, including zero. The coefficient is positive exactly at residues 0,2,5,6,8 modulo nine and negative exactly at 1,3,4,7; there are no zero or exceptional entries in the stated claim. The eta shift is -3/8 and the weight is 3/2. These shifts do not rotate the residue labels of the normalized Euler coefficients.

The formal anchor D5/S3/Arith/ModularForms/CubicThetaLattice.norm_theta_dissection is a classical lattice intermediate motivated by this question. It is not the original sign theorem. The later anchor D5/S3/Arith/ModularForms/CubicThetaCoreProduct.coeff_core_product identifies the full core lattice coefficients with the actual Euler quotient E(X^3)^3/E(X) for every natural index.

D5/S3/Arith/ModularForms/CubicThetaIdentities.B_dissection and D5/S3/Arith/ModularForms/CubicThetaIdentities.cubic_identity prove the two remaining classical Borwein identities from actual products and complete lattice sums. They are proved in Lean, not premises of the sign theorem.

D5/S3/Arith/ModularForms/EtaSignK6.F6 is the actual normalized quotient: the ninth power of Mathlib’s pentagonal Euler series times the sixth power of the unit inverse of its expansion by three. Its C6 is the coefficient at the original natural index. D5/S3/Arith/ModularForms/EtaSignK6.five_coefficient_formulas gives the two progressions modulo three and the three remaining progressions modulo nine, with factors -9, 27, 1, -6, and 9. D5/S3/Arith/ModularForms/EtaSignK6.all_index_signs proves both exact modulo-nine sign biconditionals and nonvanishing for every natural index. Ordinary partition positivity, including the empty partition, supplies strictness at every index. No separate analytic eta bridge is needed for these source-defined normalized Euler coefficients.

The supplied bounded literature review found no inspected theorem stating the complete k6 entry. This note asserts no worldwide absence, priority, unique status, independent formal acceptance, CI success, or publication of the original-problem resolution. The separate k10 entry is treated by the additional algebraic chain described below.

Citation and question boundary only; no source text or PDF is vendored.

Restricted-product auxiliaries for the separate denominator-ten row

The separate v1 Appendix A Table 1 entry 1^9 3^(-10) asks for the unshifted coefficients of E(q)^9/E(q^3)^10, including degree zero. Its proposed signs are positive at residues 0,2,5,8 and negative at 1,3,4,6,7 modulo nine, with no zero or exceptional index. This is a different endpoint from k6, notably at residue six.

The source derivation An all-index algebraic route to the denominator-ten eta signs (3 November 2026, SHA256 f9c81115730bc353681e172d5b2ad17d199d66bd4602b56a5da7dfe8e185852b) motivates new restricted-product auxiliaries. Write Q_r for the actual restricted partition series excluding residues 0,r,9-r modulo nine, and put A=Q_4, B=Q_2, C=Q_1. The new EulerNineComparison.coefficient_order proves C[n] <= B[n] <= A[n] for every natural degree by finite mixed-block cancellation and exact coefficient stabilization. Its D is the actual coloured partition product with one colour at residues three and six and two at four and five. Its multiplicity_relation proves AD=BC, and coeff_delta_nonnegative proves that AC-CD is coefficientwise nonnegative. The product (1-q)AC is also proved nonnegative, with constant coefficient one, degree-one coefficient zero, and coefficients at least one in every degree at least two.

RestrictedEulerCancellation uses Mathlib’s general restricted-partition HasProd supplier (Weiyi Wang, Apache 2.0) to prove exact finite cancellation for arbitrary allowed-part predicates and injective finite weight families. These helpers are genuinely used in the comparison proof. The new comparisons and decompositions are repository-derived implementations of the supplied source; no literature novelty is asserted for their ordinary mathematics.

Exact denominator-ten algebraic chain

LaurentSpecialization shifts Laurent row n by T^n, so the support condition k >= -n permits evaluation through ordinary polynomial coefficients. Its continuous additive map preserves complete sums and is multiplicative on supported inputs. WeightedJacobiSpecialization uses the existing Jacobi suppliers to identify every integer monomial with the limit of its actual weighted Euler products. The substitution has weights q^9 and q^r; no unrestricted Laurent substitution is used.

EulerPentagonalTrisection.euler_three_sections proves the full integer reindexing k=3m, k=1-3m, k=3m-1. The exponent and sign identities are exact and the fibres exhaust all integers. EulerNineDissection then cancels actual partition geometric factors to identify J_r=E Q_r, and proves R=E(q)/E(q^3)=A(q^3)-qB(q^3)-q^2C(q^3) in all degrees.

EulerNineSquareDissection.restricted_cubic_relation derives B^2=AC+qCD by extracting residue two from the actual cubic theta supplier and cancelling the constant-one unit A. Its positive decomposition W=(1-q)AC+q(AC-CD) proves the difficult third squared section nonnegative, with constant one, coefficient one equal to zero, and strict higher coefficients. The other sections U=A^2+2qBC and V=2AB-qC^2 have every coefficient at least one.

EtaSignK10.F10 is the actual quotient E(q)^9/E(q^3)^10, defined with the unit inverse of E(q^3); C10(n) is its unshifted coefficient. The exact first cubic lift retains factors -9 and 27. Its zero component is G10=E(q)^2/E(q^3)^3, whose sections are UP,-VP,-WP. Every ordinary partition coefficient is at least one, so smoothing removes the sole auxiliary hole, including the original residue six. EtaSignK10.all_index_signs states both complete sign biconditionals and nonvanishing for every natural n, with positive residues 0,2,5,8 and negative residues 1,3,4,6,7 modulo nine. EtaSignK10.C10_zero gives the constant one.

These are source-level mathematical anchors, not independent formal acceptance, official original-problem delivery, literature priority, unique status, required CI, or merge certification. The classical Euler/Jacobi and cubic theta mathematics is attributed through the existing suppliers; the new implementation follows the supplied algebraic manuscript and does not assert that its identities are new in the literature.