Divisor function
Named by 9 essays across 2 fields — each of them below, with the objects they name alongside it.
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.
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.
The dots a circle catches
Pick's theorem gives a lattice polygon's area exactly, with no error term anywhere. Ask a circle the same question and the exactness is gone: the count is the area plus something, the something has been measured for two centuries, and nobody knows how big it is.
Divisors that add to three times the number
The divisors of 6 add up to 12, twice 6: a perfect number. The divisors of 120 add up to 360, three times 120, and those of 30,240 to four times it. Numbers like these were a sport for Fermat and Descartes, and they are held together by one fact — the ratio σ(n)/n is a product over the primes, and each prime can add only a little.
A ratio nobody else has
Divide the sum of a number's divisors by the number and you get its abundancy: 2 for every perfect number, 12/5 for both 30 and 140. Numbers that share an abundancy are called friends. Some numbers provably have no friend at all, most have friends only far away — and for 10, whose abundancy is 9/5, nobody knows whether a friend exists.
The dots a ball catches
Count the lattice points inside a ball and compare with its volume. In five dimensions the error is about the size of one spherical shell, and that is a theorem; in four a logarithm sneaks in; in three and two the true size of the error is unknown. The difficulty runs backwards because the arithmetic of sums of squares is most erratic when there are fewest squares to add.
The product raised to the twenty-fourth power
Euler's product (1 − q)(1 − q²)(1 − q³)… has the partition numbers for its reciprocal. Raise it to the twenty-fourth power and multiply by q, and the coefficients that come out — 1, −24, 252, −1472, 4830, −6048 — multiply like the primes they are indexed by, stay inside a bound Ramanujan guessed and Deligne proved fifty-eight years later, agree with a divisor sum modulo 691, and know, modulo 23, how a cubic equation factors.
Most numbers are missing from the table
An N × N multiplication table has N² cells, and every product in it is less than or equal to N². Yet only 22.5% of the numbers that small turn up in the 10,000 × 10,000 table, and the share keeps falling. Erdős proved in 1955 that it falls to nought; Kevin Ford found in 2008 how slowly — like a tiny power of the logarithm, 0.0861 — because a product of two numbers has too many prime factors to be typical.
Divisors that take a prime whole
Six is perfect because its proper divisors 1, 2 and 3 add up to 6. Keep only the divisors that take each prime of a number completely or not at all — for 60, that is 1, 3, 4, 5, 12, 15 and 20 — and ask for the same coincidence. Below ten million it happens four times: 6, 60, 90 and 87,360. The fifth has twenty-four digits. And unlike ordinary perfect numbers, where nobody knows whether an odd one exists, here a two-line argument rules every odd number out.
Named alongside it
The objects these essays reach for when they reach for this one.
Multiplicative functionPerfect numberDivisor sumExhaustive searchPrimesAbundancePrime factorisationAsymptoticsCounting two waysError termLatticeMersenne prime