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.

Proof (PDF) · LaTeX source · Verifier

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. docs/proof.md is a shorter web outline.

Proof and verification

Component	Contents
paper/riemann.pdf (source)	Full proof and exact constants
Anthropic paper and Lean artifact	Theorem D, the optimized test family, the zero-side decomposition, and the prime-side trace estimates
docs/proof.md	Short web outline of the argument
docs/verifier.md	Interval enclosures, subdivision algorithms, and trust base
src/	Verifier source
certificates/three-point.txt	Recorded 3-point verification
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.

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:

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
• Full paper
• Lean 4 artifact
• Research conversation

License

MIT