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Open1859

Riemann Hypothesis — The Enduring Question in a Six-Page Manuscript

Do all nontrivial zeros of the zeta function have real part one-half? In a six-page manuscript of 1859, Riemann wrote that after a few unsuccessful attempts he had set aside the search for a rigorous proof. The short claim became a vast programme for measuring irregularity among the primes.

Challenge passport
First posed
1859
Time it held mathematicians
167 years open · as of 2026
Starting level
Research frontier

Small experiments are possible, but a complete proof sits at the frontier of modern mathematics.

The starting level measures how easily you can understand and test small cases. It is not the difficulty of a complete proof.

Jump to your first five minutes

Clay Mathematics Institute — $1 million (one of the seven Millennium Problems)

2026 research update · three layers of evidence

Ten trillion, 67.25%, and 100% are not one scale

All three concern the critical line Re(s)=1/2, but they quantify different logical claims.

Finite computation

First 10 trillion

Zeros already inspected

This checks an enormous initial segment. It cannot determine what one later zero must do.

Unconditional proportion theorem

≥ 67.25%

A guaranteed share of the infinite population

Previous record 5/12 ≈ 41.67%
2026 lower bound 67.25%

As the height grows, at least this proportion is guaranteed to consist of simple zeros on the critical line.

Riemann hypothesis

Every zero · 100%

A claim with no exceptions

It allows no counterexample at any height. Neither a proof nor a counterexample is known.

This is not a progress bar with “32.75% left.” It does not place the remaining zeros off the line. The new paper says the same method cannot structurally reach 100% and bears directly on neither the truth nor falsity of RH.

Increase N and watch the zeros sharpen — Riemann sandbox

Explore |ζ_N(½ + it)| through the alternating eta-series approximation. As N grows, residuals shrink at the known true zeros marked by pink dashed lines. Compare sharper approximation with its added computation cost.

0124t=0t=10t=20t=30t=40t=50Known zeros (true ζ)

An approximation of |ζ_N(½ + it)| using the alternating eta series, which converges on the critical line. As N grows, residuals shrink at the pink dashed lines marking known true zeros.

Cutoff N
200
Zeros detected
10 zeros
Mean zero residual
0.0187
Computations
72,000 ops
Notable N

Problem statement

If ζ(s) = 0 and s is a non-trivial zero, then Re(s) = 1/2. The trivial zeros are at negative even integers; the question is whether all others lie on the critical line.

The story of this puzzle

In 1859 Bernhard Riemann published a paper in the Berlin Academy proceedings connecting the count of primes with zeros of the zeta function. Its surviving manuscript is only six pages, yet it shifted attention from the average density of primes to their deviation from that average.

Why it matters: the prime number theorem gives the average trend; the Riemann hypothesis would place a strong bound on the fluctuations around it. A counterexample would not instantly break cryptography or erase number theory, but many conditional bounds would need revision. The Clay Mathematics Institute reports that the first ten trillion zeros have been checked on the critical line. This is powerful evidence, not proof about infinitely many zeros.

A new scene in 2026: an argument produced by an unreleased Claude research model and published by Anthropic raised the unconditional lower bound for simple zeros on the critical line from five-twelfths (about 41.67%) to 67.25%. It rereads mollifier calculations built across several earlier papers through a new linear-algebraic perspective; the paper, an informal note, and a Lean formalization were released together. This does not mean that “67.25% of the Riemann hypothesis is solved,” that the remaining 32.75% lies off the line, or that one merely has to fill a numerical gap to reach 100%. The paper explicitly says the method bears on neither the truth nor falsity of the hypothesis and cannot reach 100% in the same structural form.

A human aside: the atheist G. H. Hardy is said to have mailed a postcard claiming a proof before a stormy boat trip, joking that God would not let him die and leave a false theorem. It is best enjoyed as mathematical humour, not as evidence that the problem destroys minds.

Try it yourself

Mini challenge

Separate three different statements. Plot |ζ(1/2 + it)| near t ≈ 14.135, 21.022, 25.011, and 30.425 and locate the dips toward zero. Then compare: “the first N zeros lie on the line,” “at least 67.25% of all zeros are simple and on the line,” and “every nontrivial zero lies on the line.” What kind of claim is each, and why does strengthening either of the first two not automatically prove the third?

Beyond MathVoyage

Curated sources and problems. Bring one discovery back from OEIS, Project Euler, MathOverflow, or arXiv.