Authors: Keith Adler
Date: May 2026
An open-source research project testing whether ζ(3) (Apéry's constant) is algebraically independent from π, using two independent integer-relation engines: mpmath's PSLQ and LLL lattice reduction via fpylll. Every meaningful exclusion is certified by at least one engine, and cross-validated by both wherever PSLQ can reach a useful bound.
maxcoeff without also raising maxsteps (which defaults to 100 in mpmath) leaves the function returning None whether it certified anything or simply ran out of iterations - and the failure gets worse, not better, as the basis grows (0.19 certified digits/iteration on a 3-element basis collapses to 0 on a 28-element one). Three of our own results fell into this trap before we caught it - see the correction history further down this page, and paper.md §2.1c for the diagnostic and fix (LLL cross-validation) - relevant to anyone running PSLQ, not just this project. Filed upstream as mpmath/mpmath#1130.
No relation was found within the tested bounds. Strongest results: no relation a·ζ(3) + b·π² + c = 0 exists with coefficient norm up to 10¹⁹⁹⁹⁹ (LLL; independently confirmed to 10¹⁸⁶⁹⁵ by PSLQ), and ζ(3)/π³ is not algebraic of degree ≤ 100 with coefficient norm up to 10¹⁸³.
🎯 One line of results targets an open conjecture: we computed the MZV generators ζ(6,2), ζ(8,2) (depth 2, trigamma reduction) and ζ(5,3,3) (depth 3, iterated Hurwitz-tail reduction, five independent validation controls) to 1100 digits from scratch, and verified the conjectured Zagier bases at every weight through 11. Headlines: ζ(6,2) is not a rational combination of π⁸, ζ(3)²π², ζ(3)ζ(5) below coefficient norm 10²³⁷ (weight 8), and the weight-11 test passes alongside a built-in theorem-guaranteed control - adding a second depth-3 value to the conjectured 9-element basis must produce a relation (dim ≤ 9 is a theorem), and LLL finds exactly that relation with 7-digit coefficients while the conjectured basis itself shows none. All numerical evidence for the open lower-bound direction of Zagier's dimension conjecture (the upper bounds are theorems; the lower bounds are unproven - even ζ(5)/π⁵ irrational is open). Specialized MZV numerics has verified this structure to weight 22; ours is an independent reproduction within this project's certified-bound framework. See paper.md §3.9h-i, lll_zagier_check.py, lll_weight11.py.
| Result | Engine(s) | Bound |
|---|---|---|
| No relation a·ζ(3) + b·π² + c = 0 | LLL (0.2s) + PSLQ (2.9h) | 10¹⁹⁹⁹⁹ / 10¹⁸⁶⁹⁵ |
| ζ(3)/π³ not algebraic degree ≤ 100 | LLL (355s) | 10¹⁸³ |
| ζ(3)/π³ not algebraic degree ≤ 30 | LLL (7.6s) | 10⁶⁴⁰ |
| ζ(3)/π³ not algebraic degree ≤ 25 | LLL (4.4s) | 10⁷⁶⁵ |
| ζ(3)/π³ not algebraic degree ≤ 10 / ≤ 15 | LLL + PSLQ | 10¹⁸¹⁶ / 10¹²⁴⁷ |
| ζ(3) and π: no joint polynomial degree ≤ 12 | LLL (250s) | 10²⁰⁶ |
| ζ(3) and π: no joint polynomial degree ≤ 6 | LLL (5.5s) | 10⁷¹⁰ |
| ζ(3), ζ(5), ζ(7), ζ(9) linearly independent | PSLQ | 10⁸-10¹² |
| ζ(3) independent from {π, e^π, Γ(1/4)} | PSLQ | 10¹² |
| No MZV relation: ζ(3), ζ(3,2), ζ(2,3), π⁵ | PSLQ | 10¹⁰ |
| Known Li₃(1/2) identity recovered | LLL + PSLQ | validation |
LLL bounds are on the Euclidean norm ‖a‖₂ of the coefficient vector (the max-coefficient bound is smaller by at most √n, under one order of magnitude). LLL certificates rest on fplll's guarantee of LLL-reducedness plus the proven approximation factor; PSLQ certificates on mpmath's norm-bound termination. See paper.md §2 for both. Note the exclusion bound shrinks as degree grows (fixed precision spread across more basis dimensions) - the degree-100 bound is smaller than the degree-30 bound, but the degree reach is over 3× larger.
🔬 Extended sweep (§3.9f, lll_extended.py, ~11 min). Beyond the original test families, we searched five previously-untested constant families - Li₄(1/2) (whose closed form, if any, is a known open question in the literature), five more Li₃(p/q) points, new Dirichlet L-values L(χ₋₃,2/3), Γ(1/3) and Γ(1/5)-based constants, and a 10-constant "kitchen sink" joint test - and pushed the degree-30/degree-6 tests to degree 100/degree 12 to check whether the original stopping points reflected a real obstruction or just the original 2005-era PSLQ paper's practical limits (the latter: they don't). All 12 additional tests found no relation, at LLL-certified bounds from 10³⁹⁸ to 10⁹⁹⁹.
🔭 New constant families (§3.9g, lll_new_constants.py, ~71s). ζ(3) got the full algebraicity/bivariate treatment against π; Catalan's constant G and the odd zetas ζ(5), ζ(7) had only been tested for linear independence. We extended the same treatment to them (G/π² and ζ(5)/π⁵, ζ(7)/π⁷ algebraicity to degree 30; bivariate G,π to degree 8; bivariate ζ(5),ζ(7) to degree 5), plus a first look at the Euler-Mascheroni constant γ (linear vs. π,e; algebraicity to degree 5; trivariate γ,π,e to degree 6) - γ doesn't appear anywhere else in this project, and unlike ζ(3), γ's irrationality is itself an open problem. All 8 tests: no relation, bounds 10³⁵ to 10⁹⁹⁹.
🎯 Zagier dimension check (§3.9h, lll_zagier_check.py, ~7 min). Weights 0-7 re-confirm Euler's reduction theorem (single-zeta monomial counts match Zagier's d_w exactly; LLL verifies real independence). Weights 8-10 are the genuine tests: the conjectured extra generators ζ(6,2) and ζ(8,2) are depth-2 MZVs (an earlier version of this repo wrongly said depth ≥ 3 was needed), computed here to 1100 digits via the trigamma reduction ζ(a,2) = ζ(2)ζ(a) − Σψ′(m)/m^a and validated against three known closed forms before use. Results: no relation in the conjectured basis at weight 8 (bound 10²³⁷·⁷), weight 9 (10¹⁸⁹·⁸), or weight 10 (10¹³⁵·²). The weight-8 result bears on the open lower-bound direction of the conjecture. Scope caveat: specialized MZV numerics (the Blümlein-Broadhurst-Vermaseren Data Mine, 2010) verified this structure to weight 22 with dedicated tools; ours is an independent in-framework reproduction, not a frontier advance. Weight 11+ genuinely needs a depth-≥3 evaluator (depth-2 MZVs of odd weight all reduce, by Euler) - that's the flagged next step.
📜 Correction history (kept for transparency). An earlier version of this repository claimed the degree-25 (10²⁰⁰), degree-30 (10¹⁰⁰), and bivariate degree-6 (10⁵⁰) bounds from PSLQ runs that never actually reached them (mpmath's PSLQ defaults to 100 iterations - far too few). Honest PSLQ re-verification (8.7-12.3 hours per test) produced only trivial bounds (17, 2, and 0 respectively), and the claims were withdrawn - mpmath's fixed-point PSLQ simply cannot make practical progress on 26+ element bases. The LLL engine then re-established all three exclusions in seconds, at bounds hundreds of orders of magnitude beyond the original claims. Full history: paper.md §3.9.
git clone https://github.com/keithadler/zeta3.git
cd zeta3
python -m venv venv
source venv/bin/activate
pip install -r requirements.txt
python lll_tests.py # LLL suite: all headline results, <30 seconds
python run_tests.py # PSLQ suite: ~15 minutes
python lll_extended.py # degree-100/degree-12 push + 5 new constant families, ~11 min
python lll_new_constants.py # Catalan, zeta(5)/zeta(7), and gamma tests, ~71s
python lll_zagier_check.py # Zagier check incl. weight 8-10 with computed MZVs, ~7 min
python lll_weight11.py # weight-11 test with depth-3 zeta(5,3,3), ~35-40 minThe LLL suite (lll_tests.py) reproduces every headline bound in under 30 seconds. The PSLQ suite (~15 minutes, dominated by the main test's 10700 iterations) provides the independent cross-validation. The extended sweep (lll_extended.py, ~11 min) pushes the degree-30/degree-6 tests further and searches new constant families (Li4(1/2), more Li3 points, new L-values). lll_new_constants.py (~71s) extends the same algebraicity/bivariate method to Catalan's constant, zeta(5)/zeta(7), and gamma. lll_zagier_check.py (~7 min) verifies the conjectured Zagier bases through weight 10, computing the depth-2 generators zeta(6,2) and zeta(8,2). The extended PSLQ verification (40000 digits, 100000 iterations, ~2.9 hours) is documented separately in paper.md and is not part of the default suite.
├── README.md This file
├── LICENSE MIT License
├── requirements.txt Python dependencies
├── CONTRIBUTING.md How to contribute
├── paper.md Full research paper
├── lll_tests.py LLL (fpylll) test suite - all headline results, <30s
├── lll_extended.py Degree-100/degree-12 push + 5 new constant families, ~11 min
├── lll_new_constants.py Catalan, zeta(5)/zeta(7), and gamma tests, ~71s
├── lll_zagier_check.py Zagier check through weight 10 (computes zeta(6,2), zeta(8,2)), ~7 min
├── lll_weight11.py Weight-11 test with depth-3 zeta(5,3,3) + theorem control, ~35-40 min
├── run_tests.py PSLQ test suite (34 of 37 tests; 3 large-basis tests are LLL-only - see correction history above)
├── generate_figures.py Optional figure generation (requires matplotlib)
└── figures/ Generated figures for the paper
Two independent methods:
PSLQ (Ferguson-Bailey) takes real numbers computed to D digits and either finds an integer relation or certifies none exists with coefficients below a bound M, via its internal norm bound. Caveat discovered during this project: mpmath's implementation defaults to 100 iterations, and the norm bound only grows ~0.19 digits/iteration (3-element basis) - far less for large bases - so large maxcoeff values require explicitly requesting enough iterations, and 26+ element bases are impractical for it entirely.
LLL (via fpylll) reduces the lattice [I | 10^S·x] whose short vectors correspond to integer relations. fplll's guarantee of LLL-reducedness plus the proven approximation factor yields exclusion certificates directly from the shortest reduced vector - and it handles 30+ element bases in seconds.
paper.md contains the full write-up: plain-English summary, formal results, continued fraction analysis (500 terms), digit normality test, and complete appendix with all 37 test parameters.
See CONTRIBUTING.md. We especially welcome:
- Reproductions on different hardware
- Extensions to higher degree or new constant families
- Cross-validation with other software (Sage, ARPREC, Julia)
@misc{adler2026zeta3,
title={High-Precision Computational Tests on $\zeta(3)$ and $\pi$},
author={Adler, Keith},
year={2026},
note={Available at https://github.com/keithadler/zeta3}
}