Sieve — where it appears
Named by 9 essays across 3 fields — each of them below, with the objects they name alongside it.
The sieve that cannot finish
Sifting out the composites is the oldest method in the subject and it has a ceiling nobody has raised. The reciprocals of the twin primes add to a finite number, so no argument that measures thickness can reach them — and the inclusion–exclusion every sieve truncates goes wildly wrong before it goes right.
A sum stopped early still says something
Inclusion–exclusion corrects an overcount, then the correction's overcount, and so on to the end. Stop after any number of terms and the result is not merely an approximation: after an odd number it is too high and after an even number too low, always. So two or three terms bracket an answer whose full sum is out of reach — as long as the events being counted are rare.
Two halves a sieve cannot tell apart
Split the whole numbers into those with an even number of prime factors and those with an odd number. To a sieve — which sees only how many numbers each small divisor divides — the two halves look almost exactly alike. Yet sifting out the small primes leaves every prime in one half and nothing at all in the other. That is Atle Selberg's parity example, and it is the reason no sieve can prove there are infinitely many twin primes.
The patterns primes are allowed to make
Primes two apart are twin primes; primes six apart are about twice as common; three primes two apart happen exactly once, at 3, 5, 7. Which patterns of primes can recur for ever is decided by remainders alone — a pattern must leave some remainder free modulo every prime — and how often each one recurs is predicted, to within a few per cent, by a constant computed from nothing else.
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.
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.
The largest prime in a typical number
Count a number's prime factors and the answer is about log log n. Ask instead how big the largest one is, and the answer is a power of n with a law of its own: half of all large numbers have a prime factor above n^0.6065, and the share whose primes all stay below n^(1/u) is Dickman's ρ(u), a function defined by its own past. The same law governs the longest cycle of a random shuffle, and there it arrives at once; for the numbers themselves, at ten million, it is still visibly on its way.
Two neighbours share out their small primes
A number and the next one have no prime in common, and the natural guess is that their largest primes are independent. In the limit that is believed and nearly proved. Below ten million it is visibly false: neighbours are both made of small primes only nine-tenths as often as independence predicts, a run of four only two-thirds as often, and the correlation between their largest primes is negative at distance one and positive at distance six — because a small prime that divides one of two neighbours is a small prime the other cannot have.
Named alongside it
The objects these essays reach for when they reach for this one.
Prime factorisationTwin primesLargest prime factorMertens constantNormal orderPrimePrimesSmooth numberAlmost primeAlternating seriesApproximationAsymptotics