High-Precision Approximation of Riemann Zeros via the Truncated Weil Form
Abstract
The Connes-van Suijlekom truncated Weil quadratic form, indexed by a cutoff c (controlling the primes p≤ c in the operator), has a ground state whose Fourier-Mellin zeros provably lie on the critical line; whether they converge to the Riemann zeros as c→∞ is open (Connes 2026; Connes-Consani-Moscovici 2025). We present, to our knowledge, the first public implementation of the CvS Galerkin matrix at sixteen cutoffs (c=13 through $67$, plus c=100). Across c=13--$67$ at N=100, the first-zero error |γ1−γ1Riemann| shrinks monotonically from ∼ 2× 10−55 to ∼ 1.5× 10−168 (113-OOM across fifteen cutoffs). The smallest-positive even-sector eigenvalue reaches ∼ 10−334 at c=100, N=250 (275-OOM span), whose eigenvector recovers γ1,…,γ10 to 307-329 matching digits. Under the unitary equivalence with CCM 2025 Lemma 5.1, each γk is (modulo a hypothesis-status caveat at c=100) an eigenvalue of the CCM rank-one operator Dlog(λ,N). Aitken-Δ2 on the c=100 N-sweep gives log10|λ∞even|≈ −536.76 and −533.70, approaching the Connes 2026 Section 6.4 heuristic continuum prediction (≈ −530.38) monotonically in N. The empirical fit |log10λmin|≈ 13.24 c0.634 on c≤ 67, N=100 is a finite-N rate, falsified at c=100, N=200 by 49 OOM. At c=100 the raw spectrum carries a small block of negative-sign eigenvalues at the finite archimedean cutoff T=800; these are an artifact of that cutoff and absent once T is increased, so the smallest-positive even-sector eigenvalue is the genuine smallest one (continuum positivity of QWλ is RH-equivalent and is not assumed at λ=√(100)). We make no claim of proof.
Related
- More than 8/9 zeros of the Riemann zeta function are simple or on the critical line
- One-level density of zeros of the Riemann zeta function weighted by its special values
- Half-sized gaps between zeros of the Riemann zeta function
- Simple critical zeros and distinct zeros of the Riemann zeta function in short intervals
- Proportions of the non-trivial zeros of the Riemann zeta function
BibTeX
@misc{groskin2026highprecision,
title = {{High-Precision Approximation of Riemann Zeros via the Truncated Weil Form}},
author = {Akiva Groskin},
howpublished = {arXiv (Cornell University)},
year = {2026},
month = may,
doi = {10.48550/arxiv.2605.20224},
url = {https://arxiv.org/abs/2605.20224},
note = {CC BY},
}
RIS
TY - UNPB
TI - High-Precision Approximation of Riemann Zeros via the Truncated Weil Form
AU - Akiva Groskin
PY - 2026
DA - 2026/05/13
PB - arXiv (Cornell University)
DO - 10.48550/arxiv.2605.20224
UR - https://arxiv.org/abs/2605.20224
AB - The Connes-van Suijlekom truncated Weil quadratic form, indexed by a cutoff c (controlling the primes p≤ c in the operator), has a ground state whose Fourier-Mellin zeros provably lie on the critical line; whether they converge to the Riemann zeros as c→∞ is open (Connes 2026; Connes-Consani-Moscovici 2025). We present, to our knowledge, the first public implementation of the CvS Galerkin matrix at sixteen cutoffs (c=13 through $67$, plus c=100). Across c=13--$67$ at N=100, the first-zero error |γ_1−γ_1^Riemann| shrinks monotonically from ∼ 2× 10^−55 to ∼ 1.5× 10^−168 (113-OOM across fifteen cutoffs). The smallest-positive even-sector eigenvalue reaches ∼ 10^−334 at c=100, N=250 (275-OOM span), whose eigenvector recovers γ_1,…,γ_10 to 307-329 matching digits. Under the unitary equivalence with CCM 2025 Lemma 5.1, each γ_k is (modulo a hypothesis-status caveat at c=100) an eigenvalue of the CCM rank-one operator D_log^(λ,N). Aitken-Δ^2 on the c=100 N-sweep gives log_10|λ_∞^even|≈ −536.76 and −533.70, approaching the Connes 2026 Section 6.4 heuristic continuum prediction (≈ −530.38) monotonically in N. The empirical fit |log_10λ_min|≈ 13.24 c^0.634 on c≤ 67, N=100 is a finite-N rate, falsified at c=100, N=200 by 49 OOM. At c=100 the raw spectrum carries a small block of negative-sign eigenvalues at the finite archimedean cutoff T=800; these are an artifact of that cutoff and absent once T is increased, so the smallest-positive even-sector eigenvalue is the genuine smallest one (continuum positivity of QW_λ is RH-equivalent and is not assumed at λ=√(100)). We make no claim of proof.
SN - 2331-8422
ER -