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Concept

Multiplicative function

A function on the whole numbers whose value at a product of two coprime numbers is the product of its values at each. Its values are determined by what it does at prime powers, which is what turns a sum over the integers into a product over the primes.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

124361251020153060× 2 →× 3 ↑2^2 × 3 × 5 — 3 × 2 × 2 = 12 divisors

The shape of a number's divisors

Lay a number's divisors out as a lattice with one axis per prime, and two of the most useful facts in arithmetic stop being formulas and become the width and the corner of a rectangle.

number · unique factorisation 2^2 × (2^3 − 1) = 4 × 7 = 2812471428× 1× 7= 7= 49the whole rectangle adds to 7 × (1 + 7) = 56which is twice 28, so the divisors below 28 add to 28 exactly

Numbers that are their own parts

Six is one plus two plus three. Twenty-eight is one plus two plus four plus seven plus fourteen. Euclid explained where such numbers come from; Euler proved there are no others of that kind; and whether an odd one exists has been open for two thousand years.

number · perfect numbers the numbers 2, 3, 5 build12345678910111213141516171819202122232425262728293031323334353637383940414243444546474849505152535455565758596023 of the first 60100100010⁴10⁵0.00.51.01.52.02.53.0xsum of 1/p for p up to xthe sum over primesln ln x + 0.261and how slowly it divergesmultiplying the 3 truncated series together gives 3.466667, which is exactly the sum of the reciprocals of the64 numbers 2, 3, 5 can build — the marked cells, continued past 60the sum of 1/p over the primes up to x passes 2.71 at 100,000 and grows without bound, never leaving ln ln x+ 0.261 by more than 0.065

The sieve written as a product

Multiply out one geometric series for each prime and every whole number appears exactly once, as a single term. That identity turns a statement about factorisation into a statement about convergence, and it is where the analytic study of the primes begins.

number · prime distribution

Named alongside it

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

Counting two waysDivisor functionDivisor sumGeometric seriesPerfect numberPrimesUnique factorisationAbundanceConvergenceDivergenceExistence proofHarmonic series

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