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Big Numbers, Split

A hundred zeros, computedRiemann

Riemann's 1859 paper is eight pages on how many primes there are below a given size [5]. It contains a guess that has stood unproved for 167 years, a million-dollar prize, and the reason this page exists: people keep asking whether proving it would break RSA. It would not. Here is what it is, computed rather than described.

Counting primes

Write π(x) for the number of primes up to x. Gauss, at fifteen, guessed from tables that π(x) runs like x / ln x, and later that the integral li(x) = ∫ dt / ln t does better [4]. Both were proved right in 1896 [6]. The sieve in tools/compute.py counts to ten million:

xπ(x)x / ln xmissli(x)miss
10144-06.2-2.2
1022522+330.1-5.1
103168145+23177.6-9.6
1041,2291,086+1431,246.1-17.1
1059,5928,686+9069,629.8-37.8
10678,49872,382+6,11678,627.5-129.5
107664,579620,421+44,158664,918.4-339.4
Three nearly coincident lines on log axes with an inset table of misses: the count of primes, x over ln x, and li of x.
On log scales the three lines are one line — that is the Prime Number Theorem. The table is the story: li(x) misses by a few hundred where x / ln x misses by tens of thousands.

The Riemann Hypothesis is a statement about that last column. It says the miss of li(x) never grows faster than about √x · ln x — the primes are spread as evenly as a random process could spread them, with no long-range conspiracy in either direction. Every check ever made agrees; nobody has proved it.

The zeros

Riemann's function is

ζ(s) = 1 + 1/2s + 1/3s + 1/4s + … = ∏p 1 / (1 − p−s)
Reading it: the sum over all whole numbers equals a product over all primes — Euler's identity, and the reason a function about the whole numbers knows where the primes are.

Extended to complex s, it has zeros. The uninteresting ones sit at −2, −4, −6, … The interesting ones sit in the strip where the real part of s is between 0 and 1, and Riemann observed that every one he computed had real part exactly ½ — the critical line — and wrote that it was 'very probable' they all do. That sentence is the Hypothesis.

This site computed the first 100 of them, by Euler–Maclaurin summation of ζ along the line and bisection on sign changes of the Riemann–Siegel Z function, and checked the count against the Riemann–von Mangoldt formula so none was skipped. The first ten agree with Odlyzko's tables [36] to 1e-14:

#12345678910
t14.13472521.02204025.01085830.42487632.93506237.58617840.91871943.32707348.00515149.773832
A wavy teal line crossing a horizontal axis at ten marked amber points between 14 and 50.
Z(t) is ζ(½ + it) with its phase stripped off, so it is a real wave, and it crosses zero exactly where ζ does. The amber points are the first ten crossings.
A teal field with dark wells along a rose horizontal line: the size of the zeta function across the critical strip, the zeros as the dark spots.
The critical strip laid on its side: the real part of s runs down the picture from 1.4 at the top to −0.4 at the bottom, t runs left to right from 2 to 60, brightness is the size of ζ. The dark wells are the zeros, and every one sits on the rose line, where the real part is ½. Turing checked the first few thousand this way on the Manchester machine in 1950, hoping to find one off the line [10]; the count is now past ten trillion [41].

The zeros draw the primes

Here is the part that makes the Hypothesis matter. Riemann's explicit formula writes the count of primes exactly in terms of the zeros. In the cleanest form, with ψ(x) counting each prime power pk ≤ x with weight ln p:

ψ(x) = x − Σρ xρ/ρ − ln 2π − ½ ln(1 − x−2)
Reading it: the smooth guess x, minus one wave per zero ρ = ½ + iγ. Each wave has size √x and wobbles ln x · γ / 2π times per unit of ln x. Add them all and the staircase of primes comes out exactly.

The slider adds the 100 zeros one at a time. With none, the formula is the line x. With ten, the big wobbles. With a hundred, the steps at every prime and prime power are there, to within 0.37 on average between the steps.

Add the zeros

A teal staircase rising from 0 to about 94, and coloured curves that hug it more closely as more zeros are added.
The same thing drawn at build time: ψ(x) as a staircase, and the formula with 0, 10, 30 and 100 zeros.

Now the Hypothesis in one sentence: if every zero has real part ½, every wave has size exactly √x, and the primes can never stray from their smooth count by more than about √x · ln x. If some zero had real part ¾, its wave would have size x¾, and the primes would bunch and thin on a scale nobody has seen. That is all it says. It is a statement about the error term.

What it would and would not do to a key

Would not: give anyone a factoring algorithm. No method on this site — not the sieves, not the number field sieve, not Shor — runs faster if the Hypothesis is true, because none of them uses it. Their running times rest on how often values are smooth, and the heuristics for that are about smooth numbers, not zeros. A proof would be one of the great events in mathematics and would change no key size anywhere. The wallet, likewise.

Would, a little: tidy up some proofs. Miller's 1976 primality test is fast and certain if the Generalised Riemann Hypothesis holds [14]; without it, Rabin's 1980 version is fast and almost certain, and that is what every key generator on Earth uses, so nothing changes in practice. Since 2002 there has been a fast certain test with no hypothesis at all [33]. Telling prime from composite is easy. Splitting a composite is the problem, and the Hypothesis is silent on it.

Where the Hypothesis and quantum computing do meet is somewhere else entirely. In 1972 Hugh Montgomery showed Freeman Dyson his formula for how the zeros space themselves out, and Dyson recognised it on the spot: it was the spacing of the energy levels of a heavy atomic nucleus, from random matrix theory [34]. Odlyzko's computations of millions of zeros matched it to high precision [35]. The suggestion — Hilbert's and Pólya's, a century old — is that the zeros are the energy levels of some quantum system nobody has found, and Berry and Keating have a candidate [37]. A proof might come from physics. It still would not touch a key.

Recent: in 2024 Larry Guth and James Maynard gave the first improvement since 1940 on how many zeros can sit off the line in a given range [39] — progress on the Hypothesis, not a proof. Yitang Zhang's 2022 preprint on the Landau–Siegel zero, a related question, has one arXiv version and no journal [40]. Several 2025–2026 preprints claim full proofs; none has been accepted by a journal or by the field. The problem is open, and the Clay Institute's million is unclaimed [38].

The checks

25 checks run before this site publishes; 0 failing as of 2026-09-23.