Concept

Prime number theorem

The statement that the number of primes up to x is about x divided by its natural logarithm, with the ratio tending to one. Proved in 1896 by Hadamard and de la Vallée Poussin, it says the primes thin out like 1/log x.

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

The sieve of Eratosthenes below 100. A grid of the whole numbers with the composites struck out by the prime that removes them.

The primes are what is left over

Eratosthenes' sieve does not find the primes. It removes everything else, one prime at a time, and whatever survives is prime by default — which is a strange way to reach the most studied objects in arithmetic.

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

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.

Composite Counting two ways Density Discrepancy Equidistribution Farey sequence Mertens function Mobius function Prime counting function Prime gaps Primes Riemann hypothesis

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