Concept

Normal order

A function has normal order g(n) when f(n) is close to g(n) for almost every n, though not necessarily on average. The number of distinct prime factors has normal order log log n, which is the content of the Hardy–Ramanujan theorem.

Named by 2 essays across one field — 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.

number · Unique factorisation

The same bell on numbers, on primes less one, and on squares plus one. Standardised ω histograms: integers ≤ 2·10^7, p − 1 for p ≤ 2·10^7, n² + 1 for n ≤ 2·10^6.

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.

number · Unique factorisation

Named alongside it

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

Mertens constant Prime factorisation Sieve Central limit theorem Normal distribution Quadratic residue Squarefree

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