Mathematics 1859

On the Number of Primes Less Than a Given Magnitude

Bernhard Riemann

Riemann tied the scattered prime numbers to the zeros of a single complex function — and left behind the most famous unsolved problem in mathematics.

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In depth · the introduction

The prime numbers look scattered at random along the number line — and yet hidden inside them is a melody, written in a single mysterious function.

The big idea

Primes — 2, 3, 5, 7, 11, … — are the atoms of arithmetic, but they arrive with no obvious pattern. Riemann found a way to measure that pattern. He studied a function now called the Riemann zeta function, and discovered that the places where it equals zero secretly control how the primes are spread out.

Better still, he wrote down an exact recipe: a smooth curve that closely tracks how many primes there are up to any number, plus a set of fine 'waves' accounting for every wobble. The frequencies of those waves are exactly the zeros of his function. Then came the famous guess — the Riemann Hypothesis — that all of those special zeros line up perfectly along a single straight line. No one has ever proved it.

How it came about

Bernhard Riemann was a shy, sickly genius at Göttingen, a student of the great Gauss. In 1859, newly elected to the Berlin Academy, he repaid the honour with a paper — just nine pages, the only thing he ever published on the primes. In it he tossed off ideas so far ahead of their time that mathematicians are still unpacking them.

He was honest about the gap. Right after stating his hypothesis he admitted that he had tried briefly to prove it, failed, and set it aside as not essential for now. He died at 39, the proof still missing. Only decades later did others find that his private notebooks held calculations far deeper than the modest paper let on.

Why it mattered

Riemann turned a question about whole numbers into a question about a smooth function on the plane — and that translation unlocked everything. It explained why the rough rule for counting primes works, it pinned down the size of the error, and it set the agenda for a whole branch of mathematics. The hypothesis itself became the most famous unsolved problem in the field, with a million-dollar prize waiting for whoever settles it.

A way to picture it

Think of a ragged staircase that climbs one step every time you pass a prime. From far away the staircase looks like a smooth ramp — that ramp is Riemann's Li(x). But up close the steps jut above and below it. Riemann showed those jitters are a sum of pure musical tones, and the pitch of each tone is set by one zero of his function. The Hypothesis says all those tones are perfectly in tune — none louder than it should be. The primes, in other words, make a kind of music, and Riemann wrote down the score.

Where it sits

Riemann built on Euler, who first tied the primes to this sum two centuries earlier, and on Gauss, who as a boy guessed the counting law. His complex-analysis toolkit grew out of the same nineteenth-century flowering that gave us Fourier's waves (1822) — and indeed Riemann's primes, like Fourier's signals, turn out to be sums of waves. The unproven status of his hypothesis rhymes with a later discovery in this Library: Gödel's incompleteness theorems (1931) showed that some true statements may lie forever beyond proof — though whether the Riemann Hypothesis is merely hard, or genuinely unprovable, nobody knows.

The original document

Original source text

Bernhard Riemann · «Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse» · Monatsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin · November 1859

The opening

Riemann frames the paper as a gift of thanks for his election to the Berlin Academy: he will repay the honour by communicating an investigation into the frequency of the primes — a subject, he notes, that Gauss and Dirichlet had long found worthy of attention.

Meinen Dank für die Auszeichnung, welche mir die Akademie durch die Aufnahme unter ihre Correspondenten hat zu Theil werden lassen, glaube ich am besten dadurch zu erkennen zu geben, dass ich von der hierdurch erhaltenen Erlaubniss baldigst Gebrauch mache durch Mittheilung einer Untersuchung über die Häufigkeit der Primzahlen; ein Gegenstand, welcher durch das Interesse, welches Gauss und Dirichlet demselben längere Zeit geschenkt haben, einer solchen Mittheilung vielleicht nicht ganz unwerth erscheint.

The zeta function and its continuation

He begins from Euler's identity — the sum of 1/n^s over all integers equals the product of 1/(1 − p^−s) over all primes p — and takes the bold step of letting the variable s be complex. The series converges only when the real part of s exceeds 1, but Riemann continues the function ζ(s) analytically across the entire complex plane, where it is everywhere finite except for a single simple pole at s = 1.

The functional equation and the zeros

He proves a functional equation: the completed function ξ(s) = π^(−s/2) Γ(s/2) ζ(s) satisfies ξ(s) = ξ(1 − s), a mirror symmetry about the vertical line where the real part of s equals one half. This forces ζ to vanish at the negative even integers (the 'trivial' zeros) and to confine every other zero to the critical strip between 0 and 1. He estimates how many such zeros lie up to a given height.

The hypothesis

Then comes the sentence that has outlasted everything around it. Having counted the zeros, Riemann states where he believes they all lie — and, with disarming candour, sets the proof aside. (In English, after Wilkins: 'it is very probable that all roots are real. Of course one would wish here for a rigorous proof; I have for the time being, after some fleeting vain attempts, provisionally put aside the search for it, as it appears dispensable for the immediate objective of my investigation.')

… und es ist sehr wahrscheinlich, dass alle Wurzeln reell sind. Hiervon wäre allerdings ein strenger Beweis zu wünschen; ich habe indess die Aufsuchung desselben nach einigen flüchtigen vergeblichen Versuchen vorläufig bei Seite gelassen, da er für den nächsten Zweck meiner Untersuchung entbehrlich schien.

The prime count

Finally he gives an explicit formula for the number of primes below a given magnitude. Its smooth main term is the logarithmic integral Li(x); to it Riemann adds an oscillating correction for every nontrivial zero. The apparently random primes are thus expressed exactly as a smooth trend plus a sum of waves — and the frequencies of those waves are precisely the zeros of ζ(s).

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