Concept

Riemann hypothesis

The conjecture, from 1859, that every non-trivial zero of the Riemann zeta function has real part one half. It is equivalent to the primes being as evenly spread as randomness allows, and it is the most famous open problem in mathematics.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Selberg's parity example: the same sieve data, the primes on one side and nothing on the other. Horizontal bars for the numbers up to 100000 with an odd and an even count of prime factors, before sifting (50144 and 49855) and after sifting by the primes below √100000 (9527 and 0).

Two halves a sieve cannot tell apart

Split the whole numbers into those with an even number of prime factors and those with an odd number. To a sieve — which sees only how many numbers each small divisor divides — the two halves look almost exactly alike. Yet sifting out the small primes leaves every prime in one half and nothing at all in the other. That is Atle Selberg's parity example, and it is the reason no sieve can prove there are infinitely many twin primes.

number · Infinitude of primes $The 46 Farey fractions of order 12, and how far each strays from even spacing. Farey fractions of order 12 against 46 evenly spaced points, with the deviation of each drawn as a bar; largest deviation 0.0616.$

How evenly the fractions spread

List every fraction between nought and one with denominator at most n, in order. They spread across the interval almost evenly, and how fast the unevenness shrinks as n grows is — exactly, provably — the Riemann hypothesis. The link runs through a second fact: set the fractions round a circle and add them as arrows, and what is left is a whole number.

number · Stern brocot

Named alongside it

The objects these essays reach for when they reach for this one.

Almost prime Discrepancy Equidistribution Farey sequence Liouville function Mertens function Mobius function Parity Prime number theorem Primes Sieve Totient

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