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 6 essays across 3 fields — 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

The primitive 15th roots of unity, raised to a power and added. c₁₅(1) = 1; c₁₅(3) = −2; c₁₅(5) = −4; c₁₅(6) = −2; c₁₅(10) = −4; c₁₅(15) = 8.

The sums of primitive roots are always whole

Take the roots of unity of order exactly q, raise each to the power n and add them. The answer is always a whole number, it depends on n only through what n shares with q, and as n varies the sums behave like the sines and cosines of a Fourier series — so well that Ramanujan could rebuild the sum of the divisors of any number from them, and prove that their weighted total is nought, a fact that at n = 1 is the prime number theorem.

algebra · Roots of unity

Record gaps between primes against the square of the logarithm. 24 record prime gaps below thirty million as a staircase of points against log p, under the curve (log p)² and a dashed curve 1.12 (log p)²; the last record is 210 after 20831323.

The longest wait for a prime

Near a number x the primes are on average log x apart, and the longest stretch without one below x is much longer: about the square of log x, if the primes behave like random numbers with the right density. Sieving to thirty million finds twenty-four record gaps, all climbing in the shape of that square and all below it. A random model of the primes gets the scale right and the details wrong, and the correction for what it gets wrong predicts gaps larger still — larger than any anyone has found.

discrete · Prime distribution

Apéry's proof for ζ(3), as three numbers. log₁₀ of 2·lcm(1, …, n)³, of |a(n)ζ(3) − b(n)| and of their product for n up to 60: slopes 1.262, -1.546, -0.284 per step.

The race that makes ζ(3) irrational

Roger Apéry's 1978 proof that the sum of the reciprocal cubes is not a fraction comes down to a race between two numbers. A whole-number multiplier grows by a factor of ten every 0.79 steps; the gap it multiplies shrinks by a factor of ten every 0.65. The gap wins, by a margin of about seventeen per cent, and that margin is the whole proof.

number · Irrationality

Twelve's divisors make every total up to twenty-eight. Stacked bars for each total from 1 to 28, each built from distinct divisors of 12.

Divisors that make every amount

Twelve's divisors — 1, 2, 3, 4, 6, 12 — can be chosen to add to every whole number from one to twenty-eight. Numbers like that are called practical, a test on their primes decides them, and they turn out to be counted like the primes: about 1.336 x / log x of them below x. For practical numbers Goldbach's conjecture and the twin conjecture are theorems.

number · Perfect numbers

Named alongside it

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

Density Mobius function Prime gaps Primes Totient Asymptotics Composite Counting two ways Cramér's model Cyclotomic polynomial Discrepancy Divisor sum

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