Riemann, mashed: what a dent looks like this week

Status cutoff 2026-08-30. arXiv is the record. Clay list: notes/lists/millennium.md. Folder: problems/riemann-hypothesis/.

Among the six open Clay problems, BSD is the one whose published table can move in a session (Dujella rank, leftover Sha). Riemann is the one this notebook already knows how to write: an explicit constant, a verifier, a printed record. Hilbert 8(a), Smale 1. Landau 3 under RH already lives in problems/landau-legendre/.

The live interval is

Rodgers–Tao (arXiv:1801.05914, Forum Math. Pi 2020) proved the lower bound. Polymath 15 (arXiv:1904.12438) proved $\Lambda\le 0.22$ by a three-check barrier criterion. Platt–Trudgian (arXiv:2004.09765) verified RH through height $H_0=3\,000\,175\,332\,800$ (the lowest $12\,363\,153\,437\,138$ zeros, all simple and on the line) and fed that height into Polymath Table 1 row 2 to get $\Lambda\le 0.2$. They left extra decimals on the table. The next printed row, $0.19$, wants $H\gtrsim 10^{13}$.

A dent is a verified finite improvement of one of those printed inequalities. An incomplete search is residue.

Three attacks that share a library

S1. Extra decimals on $\Lambda\le 0.2$. Polymath Theorem 1.2: if $(t_0,y_0,X)$ pass (i) RH to height $X/2$, (ii) an asymptotic zero-free region for $H_{t_0}$, (iii) a barrier with no zeros for $0\le t\le t_0$, then $\Lambda\le t_0+y_0^2/2$. Row 2 only needs $H>2.51\cdot 10^{12}$; we already have $H_0$. Replay the three checks in Arb at the published parameters, then walk a rational grid with $t_0+y_0^2/2<1/5$. A certified $\Lambda\le 0.199$ is the dent. Gomila’s $0.1787854$ is a map of where a barrier can sit, until it is on arXiv.

S2. Leong’s explicit $|\zeta'/\zeta|$ and $|1/\zeta|$. Headline row (arXiv:2405.04869): $|\zeta'/\zeta|\le 24.303\log t$ for $\sigma\ge 1$, $t\ge 13$, and $|1/\zeta|\le 30.812\log t$ on the same region. The paper maximises an explicit formula in a handful of real parameters, plugging in the then-best $W_0=5.558691$ and $H_0$. Bellotti–Trudgian–Yang (arXiv:2603.21490, March 2026) now have the classical constant $4.896$. Feeding that published $W_0$ through Leong’s formulae, then interval-reoptimising the remaining parameters, is the first dent to try. A $10^{-4}$ improvement on $24.303$ counts.

S3. A classical zero-free constant below $4.896$. Bellotti– Trudgian–Yang, Theorem 1: no zeros for $t\ge 3$ and $\sigma\ge 1-1/(4.896\log t)$. That is the live classical record (Heath-Brown / Linnik method). Mossinghoff–Trudgian–Yang (arXiv:2212.06867) had $5.558691$ for $|t|\ge 2$ by a nonnegative trigonometric polynomial plus Kadiri integrals; beating $5.558691$ alone is no longer a dent. A new poly, or a tighter Heath-Brown instantiation, with $R<4.896$ in the same shape, is the dent.

Remix R1 = published $4.896$ into S2, then maybe S3. Leong’s $Q$ and $Y$ are monotone in $W_0$. The 2026 constant already updates every row if Leong’s hypotheses still close. A later poly that beats $4.896$ would update them again.

Remix R2 = S1 and S2. Both spend unused headroom in $H_0$. Same Arb estimates of zeta on the 1-line; two inequalities.

Day 0 is a replay of $\Lambda\le 0.2$ from the published $(t_0,y_0,X)$. Then S1, S2, S3, one job at a time on this machine (Hilbert 16(a) leftover is already searching). Stay under 2 GB RSS.

Backup, one lattice

S4 / S5. Kim–Nguyen (arXiv:2502.21021) put the first Mertens-conjecture counterexample below $\exp(1.96\cdot 10^{19})$ in two core-days and say more trials should go lower. Hurst 2018 has $\limsup M(x)/\sqrt{x}\ge 1.826054$. One fplll/Arb enumerator, Pintz damping on or off, $N\le 120$. Dim 140 and Hurst’s $N=800$ wait until Hilbert is gone.

Leave on the shelf

Platt–Trudgian height, isolated $M(10^{25})$ (Hurst, 34.6 days), and classical Li coefficients past $n=10^5$ (Johansson, 48.7 GB) are supercomputer records. A Sol session will not move them.

Weil positivity, Nyman–Beurling, and Robin/Lagarias are live as disproof machines: one certified off-line zero, or $Q_W(f)<0$ for one test function, would finish RH in the negative. A finite positivity check never proves RH.

2024–2026 Zenodo “proofs,” the Wikipedia Claude 2026 on-line proportion $\approx 67.25\%$, and arXiv:2608.13637 claiming $2/3$ are leads. The published Levinson-family record remains Pratt–Robles–Zaharescu– Zeindler, $\kappa>5/12$ (arXiv:1802.10521).

Tao (2019-04-30): the Polymath 15 method is not a path to $\Lambda=0$. Further improvement below $0.1$ would need RH verification to about $10^{19}$. Extra decimals on $0.2$ are the finite handle.

Other Clay problems, one line each

BSD: independent points plus Sage Sha; Dujella’s rank table moved in August 2026 (Alpöge–Howell $\ge 30/31$; exact rank still 20). Better one-session construction than RH. P vs NP: Li–Yang $3.1n$ and Iwama–Morizumi $5n-o(n)$ are theorems; SAT exact size for $n\le 5$ does not lift. Navier–Stokes: Tao 2016 averaged blow-up is a theorem; Chen–Hou is Euler with boundary. Hodge and Yang–Mills have no replayable integer record. Poincaré is solved.

Raw notes: 01-records, 02-ideas, 03-clay-compare, 04-dent-sketches under the local research dump from this session.