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High-Precision Approximation of Riemann Zeros via the Truncated Weil Form

This paper presents the first public implementation of the Connes-van Suijlekom truncated Weil quadratic form, demonstrating that its ground state eigenvalues converge monotonically to Riemann zeros with extreme precision (up to 329 digits) as the cutoff parameter increases, while empirically validating convergence rates and spectral properties without claiming a formal proof of the Riemann Hypothesis.

Original authors: Akiva Groskin

Published 2026-05-21

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Akiva Groskin

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Tuning a Radio to Find Hidden Stations

Imagine the Riemann Zeta function is a giant, complex radio station broadcasting a signal across the universe of mathematics. Hidden inside this signal are specific "frequencies" called Riemann zeros. These zeros are famous because if they all sit on a specific "critical line" (a straight line on a graph), it proves the Riemann Hypothesis, one of the biggest unsolved mysteries in math.

For decades, mathematicians have tried to find these frequencies. In 2026, a group led by Alain Connes proposed a new way to "tune" into these frequencies using a mathematical machine called the Weil Quadratic Form. They suggested that if you build this machine with a specific "cutoff" (a limit on how many prime numbers you let into the machine), the machine will produce a "ground state" (its lowest energy setting). The zeros of this ground state are supposed to line up perfectly with the Riemann zeros.

The Big Question: Does this machine actually work? As you make the machine bigger and let in more prime numbers (increasing the cutoff), do the zeros get closer and closer to the real Riemann zeros, or do they just wander around?

What This Paper Did: Building the Machine from Scratch

The author, Akiva Groskin, didn't just guess; they built the machine from scratch using a computer.

Independent Reproduction: No one else had publicly shared the code to build this specific machine. Groskin wrote their own code (in Python) to recreate the "Connes–van Suijlekom" matrix. It's like building a replica of a famous, complex clockwork engine from a blueprint to see if it actually ticks.
The Test: They ran the machine at 16 different settings (cutoffs), starting small (letting in primes up to 13) and getting larger (up to 100).
The Result: The machine worked incredibly well.
- At the smallest setting, the error was tiny (about 1 part in $10^{55}$ ).
- At the largest setting (cutoff 67), the error became unimaginably small (about 1 part in $10^{168}$ ).
- Analogy: Imagine trying to hit a bullseye on a target the size of a grain of sand from across the galaxy. At the start, you were off by a few inches. By the end, you were off by less than the width of a single atom.

The "C = 100" Experiment: Pushing the Limits

The paper didn't stop at the standard settings. They pushed the machine to a much larger setting: cutoff 100. This is like turning the volume up to maximum to see if the signal stays clear or gets distorted.

The Challenge: At this high setting, the machine started showing some "static" (mathematically, some negative numbers appeared where there should only be positive ones). This is like a radio picking up a bit of static noise.
The Solution: The author carefully filtered out the static and focused on the "positive branch" of the signal.
The Discovery: Even with the static, the machine was able to predict the first 10 Riemann zeros with 300+ correct digits. This is the deepest, most accurate prediction of its kind ever made publicly.
The Comparison: They compared their results to a "heuristic prediction" (a smart guess) made by Connes. Their results matched the guess very closely, suggesting the machine is heading in the right direction, though they didn't prove the guess is a law of nature.

Key Findings and "Structural" Observations

The paper is full of interesting patterns they found while building the machine:

The "Shape" Stays the Same: Even though the "volume" (the eigenvalues) changed by massive amounts (trillions of times smaller), the "shape" of the signal (the eigenvector) stayed almost identical. It's like a guitar string that gets quieter and quieter but keeps vibrating in the exact same pattern.
It's Not Just About Primes: They found that the improvement in accuracy came mostly from making the "room" (the interval) bigger, not just from adding more prime numbers. It's the size of the container that matters most.
No Simple Formula: They tried to fit the data to simple mathematical curves (like a straight line or a smooth curve), but none of them worked perfectly. The convergence (how fast it gets better) is complex and doesn't follow a simple rule yet.
Randomness vs. Order: The "bulk" of the machine's internal numbers (the ones that aren't the main zeros) behaved like random, unconnected events (Poisson statistics), rather than the highly ordered, repelling pattern (GUE statistics) that the actual Riemann zeros are famous for. This suggests the "noise" in the machine is different from the "signal" we are looking for.

What They Did NOT Claim (The "Honest Non-Claims")

The author is very careful not to overpromise:

No Proof of the Riemann Hypothesis: They did not prove the hypothesis is true. They just showed that this specific machine gets incredibly close to the answer.
No Convergence Proof: They didn't prove that the machine will eventually hit the exact answer as it gets infinitely big. They just showed it gets closer and closer in the range they tested.
No Magic Bullet: The "negative numbers" (static) that appeared at the high setting are still a mystery. They might be a glitch in the math, or they might be a real feature. The author didn't solve this; they just reported it.

The Bottom Line

This paper is a massive experimental mathematics achievement. The author built a complex mathematical engine, tuned it with extreme precision, and showed that it can predict the Riemann zeros with unprecedented accuracy.

Think of it as a master watchmaker building a new type of clock. They didn't prove that time works the way Einstein said it does, but they built a clock that keeps time so perfectly that it matches the theoretical predictions of time almost exactly. This gives other mathematicians a powerful new tool to study the mystery, even if the final "proof" is still out of reach.

The paper concludes with a promise: All the code, data, and results are public. Anyone can download the "machine," run it, and verify the results themselves.

Technical Summary: High-Precision Approximation of Riemann Zeros via the Truncated Weil Form

Problem Statement
The paper addresses an open question within the spectral interpretation of the Riemann Hypothesis (RH) proposed by Alain Connes and collaborators. Specifically, it investigates the convergence of the ground state zeros of the truncated Weil quadratic form to the actual Riemann zeros as the cutoff parameter $c$ (controlling primes $p \le c$ ) tends to infinity. While Connes–van Suijlekom (2025) proved that these zeros lie on the critical line for any finite $c$ , the convergence to the true Riemann zeros as $c \to \infty$ remains unproven. Prior to this work, no independent public implementation of the Connes–van Suijlekom Galerkin matrix existed, and no independent reproduction of the numerical data reported by Connes (2026) or Connes–Consani–Moscovici (2025) had been published.

Methodology
The author presents a high-precision numerical implementation of the Connes–van Suijlekom Proposition 4.1 Galerkin matrix using Python and the mpmath library for arbitrary-precision arithmetic.

Operator Construction: The implementation constructs the even-sector projection of the truncated Weil quadratic form on $L^2([0, \log c])$ . The matrix entries are derived from a kernel involving the archimedean Mellin multiplier (involving the digamma function) and a finite sum over primes $p \le c$ .
Unitary Equivalence: The paper establishes that the implemented matrix is unitarily equivalent to the matrix defined in Connes–Consani–Moscovici (2025) Lemma 5.1, confirming that the computation targets the same spectral object.
Numerical Strategy:
- Precision: Computations utilize up to 1000 decimal digits (dps) to avoid backward-error floors, with specific sweeps at dps=500 and dps=1000.
- Quadrature: Adaptive tanh–sinh quadrature is used for the archimedean integral, with a cutoff $T=800$ verified for convergence.
- Eigenvalue Extraction: The smallest positive eigenvalue ( $\lambda_{\min}^{even}$ ) of the even-sector matrix is computed. The corresponding eigenvector is used to extract Riemann zeros ( $\gamma_k$ ) via the Fourier–Mellin transform of the ground state.
- Validation: The study includes cross-validation against published values at $c=13$ and $c=14$ , and an out-of-sample test at $c=100$ with a sweep of basis sizes $N \in \{100, 150, 200, 250\}$ .

Key Contributions

First Independent Implementation: This is the first public, independent reproduction of the Connes–van Suijlekom Galerkin matrix. It reproduces the $c=13$ first-zero error within a factor of 1.3 of the values reported by Connes (2026) and Connes–Consani–Moscovici (2025).
Extended Numerical Sweep: The study extends measurements to fifteen cutoffs ( $c=13$ through $67$) and a sixteenth cutoff at $c=100$ . Across the in-sample window ( $c \le 67$ ) at $N=100$ , the absolute error $|\gamma_1 - \gamma_{\text{Riemann}}|$ decreases monotonically from $\sim 2 \times 10^{-55}$ to $\sim 1.5 \times 10^{-168}$ , spanning 113 orders of magnitude.
Out-of-Sample Test at $c=100$ : At $c=100$ , the author performs an $N$ -sweep to test convergence behavior. Using Aitken- $\Delta^2$ extrapolation on the sequence of smallest-positive eigenvalues, the estimated limit $\log_{10}|\lambda_{\min}^{even}(\infty)|$ is found to be approximately $-533.7$ to $-536.8$. This approaches the heuristic continuum prediction of Connes (2026) ( $\approx -530.4$ ) with gaps of 3.3 to 6.4 orders of magnitude, providing empirical consistency with the theoretical heuristic.
Structural Characterization: The paper identifies several structural properties of the finite- $N$ $N$ operator:
- Eigenvector Invariance: Despite eigenvalues varying by 113 orders of magnitude, the ground-state eigenvectors across different cutoffs exhibit high overlap ( $>0.95$ ), suggesting approximate $c$ -invariance.
- Convergence Rates: All ten detectable zeros ( $\gamma_1$ through $\gamma_{10}$ ) converge at rates within 3.8% of each other.
- Sobolev Regularity: An empirical Galerkin convergence exponent $s(c) \approx 55 \log c - 128$ is identified, suggesting increasing regularity with $c$ .
- Bulk Statistics: The bulk spectrum (excluding the ground state) follows Poisson statistics ( $\beta < 0.05$ ), indicating no level repulsion, which the authors note is a structural diagnostic of the truncated operator rather than a test of Montgomery's conjecture (which applies to locally rescaled zeros).
Extension to Dirichlet L-functions: A preliminary port to $L(s, \chi_3)$ reveals an oscillating positivity breakdown at specific cutoffs ( $c=23, 29$ ), contrasting with the stable positivity observed in the Riemann zeta case for $c \le 67$ .

Results and Data

Error Reduction: At $c=100, N=250, \text{dps}=500$ , the extraction of the first ten Riemann zeros achieves 307–329 matching digits.
Falsification of Simple Models: Eight two-parameter smooth convergence models (linear, quadratic, exponential in $L$ or $\pi(c)$ ) were tested and rejected. A five-parameter log-periodic model was also falsified out-of-sample at $c=53$ .
Finite-N vs. Continuum: The empirical fit $|\log_{10} \lambda| \approx 13.24 c^{0.634}$ derived from $c \le 67$ data is shown to be a finite- $N$ rate, not the continuum asymptote. It is falsified at $c=100$ by a 49-order-of-magnitude deviation in the direction of faster decay.
Negative Eigenvalues: At $c=100$ , the raw finite- $N$ matrix contains a small block of negative-sign eigenvalues. The author reports the smallest positive eigenvalue as an empirically stable branch, noting that the identification of this branch with the theoretical ground state is conditional on the nature of these negative values (likely condition-driven artifacts).

Claims and Significance
The paper explicitly does not claim to prove the Riemann Hypothesis or validate Connes' convergence conjecture.

Nature of Contribution: The work is presented as "experimental mathematics" providing reproducible numerical data and careful interpretation under the existing Connes–van Suijlekom/Connes–Consani–Moscovici framework.
Consistency, Not Proof: The agreement between the Aitken extrapolation at $c=100$ and the Connes (2026) heuristic is reported as an empirical consistency test, not a theorem. The author emphasizes that Aitken acceleration relies on assumptions about the local form of the convergence sequence.
Hypothesis Status: The application of Connes–Consani–Moscovici Theorem 1.1 to the $c=100$ data is treated with a "hypothesis-status caveat" due to the presence of negative eigenvalues in the raw matrix, which are not assumed to exist in the continuum limit.
Significance: The primary significance lies in providing the first independent, high-precision numerical verification of the Connes–van Suijlekom framework, extending the range of tested cutoffs, and characterizing the structural behavior (eigenvector stability, convergence rates, and spectral statistics) of the truncated operator. The data constrains theoretical work by quantifying the finite- $N$ effects and demonstrating that simple smooth models cannot capture the convergence behavior over the measured range.