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zeta-simple-zeros

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Reproducible verification for a 67.30085% lower bound on simple zeta zeros on the critical line

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# A 67.30085% lower bound for simple zeros of the Riemann zeta function

This repository gives a proof and reproducible finite verification for
an extension of Theorem D in Claude's paper
[*More Than Two Thirds of the Zeros of the Riemann Zeta Function Lie on the Critical Line*](https://www-cdn.anthropic.com/564f962e60643842f5fcb4a17c9dbc8f608f1c37.pdf).

**[Proof (PDF)](paper/riemann.pdf)** · [LaTeX source](paper/riemann.tex) · [Verifier](docs/verifier.md)

Let $N(T,2T)$ count zeros with multiplicity and let
$N_0^s(T,2T)$ count simple zeros on the critical line. The main result is

$$
\liminf_{T\to\infty}\frac{N_0^s(T,2T)}{N(T,2T)}
\ge 0.673008527927\ldots.
$$

## Results

Anthropic's Theorem D proves

$$
\liminf_{T\to\infty}\frac{N_0^s(T,2T)}{N(T,2T)}
\ge
\frac32-\frac1{\sqrt2}\cot\frac1{\sqrt2}
=0.672500703679\ldots.
$$

The additional argument in this repository gives two explicit improvements:

| Argument | Certified inequality | Lower bound |
| --- | ---: | ---: |
| 3 consecutive zeros | $\epsilon_4\ge 221/10^6$ | **67.2519767%** |
| 7 consecutive zeros | six-variable bound $\ge 19/5000$ | **67.3008528%** |

Both use the analytic estimates and the optimized test family from the
Anthropic paper.

## Argument

The rank–trace step in the paper uses the rank, inertia, and two traces of a
Hermitian matrix. Its equality case permits the vectors associated with simple
zeros to be mutually orthogonal. For the vectors produced by the optimized
test family, however, each inner product is determined by the difference
between two zero ordinates.

The proof keeps this extra information through a stability refinement of the
rank–trace inequality.

### 3 consecutive zeros

If the consecutive gaps are $u$ and $v$, the three pairwise differences
are $u$, $v$, and $u+v$. The Montgomery–Taylor kernel cannot vanish at
all three values. The verifier makes this quantitative on $u+v\le4$, and the
resulting local estimate yields the 67.2519767% bound.

### 7 consecutive zeros

Seven consecutive zeros have 21 pairwise differences determined by six gaps.
The verifier proves a weighted lower bound for all six nonnegative gaps using
Arb interval arithmetic and exhaustive subdivision. Averaging this estimate
over consecutive blocks yields the 67.3008528% bound.

The full proof is in [`paper/riemann.pdf`](paper/riemann.pdf).
[`docs/proof.md`](docs/proof.md) is a shorter web outline.

## Proof and verification

| Component | Contents |
| --- | --- |
| [`paper/riemann.pdf`](paper/riemann.pdf) ([source](paper/riemann.tex)) | Full proof and exact constants |
| Anthropic paper and [Lean artifact](https://github.com/anthropics/zeta-23-lean) | Theorem D, the optimized test family, the zero-side decomposition, and the prime-side trace estimates |
| [`docs/proof.md`](docs/proof.md) | Short web outline of the argument |
| [`docs/verifier.md`](docs/verifier.md) | Interval enclosures, subdivision algorithms, and trust base |
| [`src/`](src/) | Verifier source |
| [`certificates/three-point.txt`](certificates/three-point.txt) | Recorded 3-point verification |
| [`certificates/seven-point.txt`](certificates/seven-point.txt) | Recorded 7-point verification |

The verifier checks the two finite inequalities used by the new argument. It
reconstructs every transcendental enclosure from the formulas on each run.

## Run the verifier

Python 3.9 or later is required.

```bash
python3 -m venv .venv
source .venv/bin/activate
python3 -m pip install --upgrade pip
python3 -m pip install -e .

# Fast 3-point verification
zeta-zero-verify three

# Exhaustive 7-point verification; allow a few minutes
zeta-zero-verify seven --progress-every 1000000
```

Run the tests with:

```bash
python3 -m unittest discover -s tests -v
```

## Status

Research draft generated by GPT-5.6 Sol. Independent verification and peer
review are welcome.

## References

- [Anthropic research article](https://www.anthropic.com/research/riemann-zeta)
- [Full paper](https://www-cdn.anthropic.com/564f962e60643842f5fcb4a17c9dbc8f608f1c37.pdf)
- [Lean 4 artifact](https://github.com/anthropics/zeta-23-lean)
- [Research conversation](https://chatgpt.com/share/6a7a7ff8-1c88-83e8-88d1-021a1bbdc8fe)

## License

MIT

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