Mertens constant
Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.
How many primes a typical number has
A typical number near N has about log log N different prime factors, and the count is spread around that in a bell curve whose variance is log log N as well. Both facts are theorems. Neither is visible at any size anyone can count: up to ten million the average is right and the spread is less than half what the limit says.
The same bell on thinner sets
A typical whole number has about log log N prime factors, spread in a bell curve. That was a theorem about all numbers, built on the picture of each prime dividing independently. The numbers one less than a prime, and the numbers n² + 1, are far too thin for that picture to have any obvious claim on them, and on both the same bell appears — with means shifted by constants that come straight from how often each small prime divides them. The squares plus one that are prime, the case of a single factor, number 102,205 up to n = two million against 102,302 predicted; whether there are infinitely many is open.
A share that depends on the average
What share of the whole numbers begin with the digit 1? Counted up to N, the answer swings between one ninth and five ninths for ever, and averaging the swing over N, even three times over, narrows it without removing it. Weight each number n by 1/n instead and the share settles at log₁₀ 2 — Benford's value — and so does every other sensible weighting that counts each decade alike. The primes behave the same way. Benford's law for the counting numbers is not a fact about them; it is a fact about how they are averaged.
Named alongside it
The objects these essays reach for when they reach for this one.
Normal orderPrime factorisationSieveCentral limit theoremConvergence rateEquidistributionLimitLogarithmNormal distributionPrime number theoremQuadratic residueSquarefree