GoKawiilThe primes 2, 3, 5, 7, 11, 13, … are the atoms of arithmetic: every whole number factors into primes in exactly one way, so facts about primes become facts about all numbers. Individually they are irregular — no known rule produces the next prime from the ones before it. Counted in bulk, they obey a law: the number of primes up to x stays close to a single smooth curve (the prime number theorem, proved in 1896). The open question is the size of the error — how far the true count can stray from the curve. That error term is what the Riemann hypothesis governs, and it is why RH matters: sharpen the error term and you sharpen hundreds of results in number theory that depend on it.
In 1859 Bernhard Riemann explained where that hidden order comes from. He took Euler's identity, which connects the primes to a single function of one complex variable,
ζ ( s ) = ∑ n ≥ 1 1 n s = ∏ p p r i m e ( 1 − p − s ) − 1 \htmlData{term=zeta, tc=v}{\zeta(s)}\;=\;\htmlData{term=sum, tc=s}{\sum_{n\ge1}\frac{1}{n^{s}}}\;=\;\htmlData{term=prod, tc=b}{\prod_{p\ \mathrm{prime}}\left(1-p^{-s}\right)^{-1}} ζ ( s ) = n ≥ 1 ∑ n s 1 = p prime ∏ ( 1 − p − s ) − 1 hover or tap a colored term for what it does
extended it to the whole complex plane, and discovered that the wobble of the prime count around its smooth curve is governed — exactly, via an explicit formula — by the locations of the zeros of this function. Each zero contributes one wave to the error; the zero's height sets the wave's frequency and, crucially, its horizontal position sets the wave's amplitude. Riemann observed that every zero he could examine sat on one vertical line, Re s = ½, now called the critical line — the position giving the smallest possible amplitude — and remarked it was “very probable” all of them do. That remark is the Riemann hypothesis. Its concrete content: the prime-count error up to x never exceeds roughly √x, the same size as the wobble of a fair coin flipped x times. The primes are allowed to look random; RH says they are never allowed to drift with a bias.
The wave description is an actual formula, and you can run it below. The slate staircase counts prime powers (a cousin of the staircase above, weighted so the mathematics is exact), and the vermillion curve is Riemann's formula built from the smooth trend plus one wave per zeta zero. Drag the slider and watch thirty zeros carve the primes: