The Riemann Hypothesis

1Counting primes, and a smooth average

Write $\pi(x)$ for the number of primes up to $x$. This $\pi$ is not 3.14; it is the prime-counting function. For example $\pi(10) = 4$ (the primes 2, 3, 5, 7), $\pi(100) = 25$ and $\pi(1000) = 168$. Its graph is a staircase that steps up by one at each prime.

The Prime Number Theorem, proved in 1896 by Hadamard and de la Vallée Poussin, says the staircase follows a smooth curve. Near $x$, the chance a number is prime is about $1/\ln x$, where $\ln$ is the natural logarithm. Add up those chances and you get the logarithmic integral $\operatorname{Li}(x)$.

At $x = 1000$ the true count is 168. The estimate $\operatorname{li}(x)$ gives about 177.6, and a better curve due to Riemann, $R(x)$, gives about 168.36. The average is under control. What the theorem does not tell you is how far the staircase can wander from the average.

2Zeta knows the primes

Euler's product shows that zeta is built from the primes: one factor per prime. So anything you learn about zeta is secretly a fact about primes. In particular, the places where zeta is zero turn out to control how the primes are spread.

The sum $\sum 1/n^s$ only makes sense when the real part of $s$ is more than 1. Riemann's first great step, in his 8-page paper of 1859, was to extend zeta smoothly to every complex number except $s = 1$. Once you do that, zeta has zeros. Some are trivial, at $-2, -4, -6, \dots$ and fully understood. The rest are the mystery.

3The zeros and the critical line

Every nontrivial zero $\rho = \beta + i\gamma$ is known to lie in the critical strip $0 < \beta < 1$. Here $\beta$ is the zero's real part and $\gamma$ its height. Zeros come in mirror pairs: if $\tfrac12 + 14.13i$ is a zero, so is $\tfrac12 - 14.13i$.

Riemann remarked that it is "very probable" that all of them lie on the middle line $\beta = \tfrac12$. That remark is the Riemann Hypothesis. The first few heights are 14.134725, 21.022040, 25.010858, 30.424876 and 32.935062, and each of them sits on the line.

Clay's official page says the first $10^{13}$ (ten trillion) zeros have been checked numerically. A fully rigorous computation by Platt and Trudgian confirms every zero up to height $3 \times 10^{12}$, which is about $1.24 \times 10^{13}$ zeros. Not one has ever been found off the line.

4Riemann's explicit formula, the staircase made of waves

Riemann wrote the exact prime count as a smooth curve minus one correction for each zero. That is what the widget above computes. $R(x)$ is the smooth part. Each zero $\rho$ adds a wave $R(x^\rho)$, and pairing each zero with its mirror twin makes the wave real.

A zero at height $\gamma$ makes a wave that wiggles $\gamma$ times faster in $\ln x$. Low zeros paint the broad shape; high zeros add the sharp corners. With all infinitely many zeros, the formula gives the staircase exactly, taking the half-way value at each prime.

A cleaner version uses Chebyshev's weighted count $\psi(x)$, where each prime power $p^k \le x$ adds $\ln p$. There, each zero contributes $x^\rho/\rho$. The size of $x^\rho$ is $x^\beta$, so a zero's real part sets how loud its wave is.

5Why one half means the primes are tame

If every zero has real part $\tfrac12$, every wave has size about $x^{1/2} = \sqrt{x}$. That is as quiet as the waves can possibly be. If even one zero had real part $0.8$, its wave would grow like $x^{0.8}$ and drown out the rest, and the primes would be far more irregular than we believe.

In 1901 von Koch made this exact: the Riemann Hypothesis is true precisely when the error in counting primes is no bigger than a constant times $\sqrt{x}\,\ln x$. Schoenfeld (1976) worked out the constant, assuming RH, for all $x \ge 2657$.

So the hypothesis is not only about zeros. It is a precise promise that the primes stray from their average by about $\sqrt{x}$ and never more. The "one half" in $\operatorname{Re}(\rho) = \tfrac12$ becomes the square root in the primes.

6How much do we know?

Since nobody can prove all zeros are on the line, mathematicians prove that many are. Hardy (1914) showed infinitely many are. Selberg (1942) got a small positive proportion. Levinson (1974) reached at least one third, and Conrey (1989) at least two fifths.

The best accepted, peer-reviewed bound is due to Pratt, Robles, Zaharescu and Zeindler: more than $5/12$, about 41.7%, of zeros are on the line (posted 2018, published 2020). In 2024 Guth and Maynard proved a new limit on how many zeros could be off the line far from the middle, the first big improvement on Ingham's bound from 1940. Experts called it sensational, but it does not prove RH.

Even 100% would not be enough. A proportion of 100% can still leave infinitely many exceptions, because there are infinitely many zeros.

7The 2026 AI claims, said exactly

On 10 August 2026 Anthropic claimed that an internal Claude model produced a paper raising the proportion of zeros known to be on the line to at least 67.2%. A Lean formalization (a computer-checked proof) passes Lean's checker, to the extent its statement matches the theorem. Outside experts were shown it, but they have not publicly endorsed it, and it is not peer reviewed. Anthropic itself said it does not expect the method to lead to a proof of RH.

On 6 October 2026 OpenAI released a claimed AI-generated proof of the "quasi-Riemann hypothesis": that every nontrivial zero has real part between $\tfrac{1}{12}$ and $\tfrac{11}{12}$. Per Quanta (19 October 2026), experts largely think it is correct, but it has not yet been checked in full, and the $\tfrac{11}{12}$ version is not Lean-verified (an earlier, longer version with the strip from $\tfrac18$ to $\tfrac78$ is).

Neither claim is the Riemann Hypothesis. One says "at least two thirds of the zeros are on the line"; the other says "no zero is very close to the edges of the strip". RH says every zero is exactly on the middle line. If both claims hold up, they are real progress. They would not settle the problem.