files / problems / riemann-hypothesis / PROBLEM.md

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Riemann hypothesis

In general

The Riemann zeta function is initially

ζ(s)=∑n≥1n−s\zeta(s)=\sum_{n\geq1}n^{-s}

for ℜs>1\Re s>1 and continues meromorphically to the complex plane. Its nontrivial zeros lie in 0<ℜs<10<\Re s<1. The Riemann hypothesis says that every one of them has real part 1/21/2. Clay lists the problem as open.

This folder uses the de Bruijn–Newman constant Λ\Lambda as its finite handle. The heat-deformed Riemann Ξ\Xi function has only real zeros exactly for t≥Λt\geq\Lambda. Riemann's hypothesis is equivalent to Λ≤0\Lambda\leq0, while Rodgers–Tao proved Λ≥0\Lambda\geq0. Polymath 15 proved an effective upper-bound criterion and Λ≤0.22\Lambda\leq0.22; Platt–Trudgian's verification of RH through height 3,000,175,332,8003{,}000{,}175{,}332{,}800 supplies the published instantiation Λ≤0.2\Lambda\leq0.2.

Landau 3 is kept separately in problems/landau-legendre/. Its exponent 0.225250.22525 is conditional on RH and does not change the Clay problem.

Precise finite question

Polymath 15 Theorem 1.2 gives Λ≤t0+y02/2\Lambda\leq t_0+y_0^2/2 when three inputs are certified: RH through X/2X/2, an asymptotic zero-free region at time t0t_0, and a zero-free barrier over 0≤t≤t00\leq t\leq t_0. The published explicit endpoint to beat is therefore

Λ≤15.\Lambda\leq\frac15.

The first computation asks whether a complete, independently replayable instantiation makes the right side strictly smaller than 1/51/5. It also replays a published Lehmer-pair lower-bound calculation as a check on the opposite finite handle.

What would count as a new bound

A dent is a checkable inequality Λ≤c<0.2\Lambda\leq c<0.2 whose zero-height, finite, tail, barrier, and analytic inputs have all been checked against published theorems, with independent finite verification. Arithmetic on rounded table entries is insufficient. An off-arXiv target is credited as an earlier candidate; its complete certificate and analytic transfer must be independently checked before this folder asserts it.

Rodgers–Tao already give the exact lower endpoint Λ≥0\Lambda\geq0. A stricter lower bound would contradict RH, so replaying an older negative Lehmer-pair bound cannot move the present record.

Current result

The complete certificate in compute/q2/CLAIM.md establishes

Λ≤8939275000000=0.1787854<15.\Lambda\leq\frac{893927}{5000000}=0.1787854<\frac15.

The candidate value and construction are due to Jude Gomila. This notebook freshly regenerated all 3,149,013 finite rows, checked the complete range with a new Rust interpolation algorithm, and reproduced the height, error, tail, and full 883-prism barrier computations. The analytic review includes the closed zero-time limit and the normalization of the finite sums. Both full finite algorithms give floor 791366/1012791366/10^{12}; subtracting the uniform error leaves the exact positive margin 557871094787662151/1024557871094787662151/10^{24}.

The comparison is with Platt–Trudgian's published 1/51/5 bound. The independent algorithms share Arb, and the mathematical review is by GPT-6 Astra, not an external human referee. The proof and these independence boundaries are recorded in compute/q2/NOTE.md. RH remains open.

Earlier checks

The rounded Polymath Table 1 parameters (t0,y0)=(0.186,0.16733)(t_0,y_0)=(0.186,0.16733) satisfy

t0+y022=0.19999966445,t_0+\frac{y_0^2}{2}=0.19999966445,

but those printed decimals do not certify the extra digits. The published theorem remains Λ≤0.2\Lambda\leq0.2.

In q1, the off-arXiv 0.17878540.1787854 candidate passed its stored assembly review here. Fresh Arb computations reproduced its finite-error majorant, infinite tail, and 883-prism barrier at two precisions where applicable. The stored finite range of 3,149,013 rows was parsed, and a fresh producer run matched its first row; the remaining rows were not regenerated. The analytic bridge was not independently reviewed. That campaign ended as residue. The exact small checks and retained fresh logs are in compute/q1/.

Primary sources