🪴 GoDeep
Search
← University Challenge
Portrait of The Riemann hypothesis

Science · Mathematics

The Riemann hypothesis

1859

Mathematics — analytic number theory

The conjecture that every non-trivial zero of the Riemann zeta function has real part exactly one half — that they all lie on a single vertical "critical line" in the complex plane. Bernhard Riemann proposed it almost in passing in an eight-page paper of 1859, his only publication on number theory, while investigating how the prime numbers are distributed. It was the eighth of Hilbert's problems in 1900 and is one of the seven Millennium Prize Problems set by the Clay Mathematics Institute in 2000; it remains open, and is widely regarded as the greatest unsolved problem in mathematics.

How to spot them

  • ▸All non-trivial zeros lie on the critical line, real part one half
  • ▸Proposed in an eight-page paper of 1859, its author's only work on number theory
  • ▸Eighth on Hilbert's 1900 list and worth a million dollars from the Clay Institute
  • ▸Would pin down the error term in the count of primes below a given number
  • ▸Hundreds of published theorems begin 'assuming the hypothesis…'

Key facts

An eight-page 1859 paper 'On the number of primes less than a given magnitude' (1859)

Its author's only paper on number theory, read to the Berlin Academy

The eighth of the 23 problems posed in Paris in 1900 (1900)

David Hilbert's list at the International Congress of Mathematicians

The claim that every non-trivial zero has real part exactly one half

The statement itself: all the zeros sit on the critical line

The conjecture about zeta zeros that would sharpen the prime number theorem's error term

Its chief consequence: the tightest possible control on how the primes are spaced

Hilbert's remark that on waking after 500 years his first question would be whether it had been solved

A much-quoted story about the problem's standing

Hardy's 1914 proof that infinitely many zeros lie on the line, though not that all do (1914)

G. H. Hardy; about 40% are now known to lie on it

The zeros' spacing statistics, which match those of energy levels in heavy atomic nuclei (1972)

The Montgomery–Dyson encounter over tea at Princeton, linking it to random matrix theory

Key ideas

  • Euler's product formula expresses the zeta function as a product over all primes, which is why a statement about its zeros is a statement about the primes.
  • The prime number theorem says the number of primes below n is roughly n/ln n. The hypothesis would give the sharpest possible bound on how far that estimate can stray.
  • The 'trivial' zeros are at the negative even integers, −2, −4, −6 and so on; the hypothesis concerns all the others.

Quick-fire facts

  • •Bernhard Riemann (1826–1866) published the conjecture in November 1859 and died of tuberculosis at 39 in Italy
  • •The trivial zeros are the negative even integers; the hypothesis concerns the non-trivial ones
  • •It is Hilbert's eighth problem, posed in Paris in 1900, and one of the seven Clay Millennium Prize Problems announced in 2000
  • •Only one Millennium problem has been settled: the Poincaré conjecture, by Grigori Perelman, who declined the million dollars in 2010
  • •G. H. Hardy proved in 1914 that infinitely many zeros lie on the critical line
  • •More than ten trillion zeros have been checked by computer and all lie on the line
  • •Hugh Montgomery's 1972 conversation with Freeman Dyson revealed that the statistics of the zeros match those of eigenvalues of random Hermitian matrices
  • •Louis de Branges has repeatedly claimed proofs, none accepted; Michael Atiyah announced one in 2018 that was not accepted either
  • •An equivalent statement is a bound on the Mertens function, and another involves the divisor-sum function and the harmonic numbers
  • •The analogue for function fields over finite fields was proved by André Weil, and the Weil conjectures were completed by Pierre Deligne in 1974
  • •Marcus du Sautoy's The Music of the Primes (2003) and John Derbyshire's Prime Obsession (2003) are the standard popular accounts

Drill it — 8 cards

1 / 8new card

Sources