One of the undetermined numbers in Les nombres remarquables is the Mills number (or numbers; this is not uniquely defined, as will be clear from the description below). I had somehow forgotten all about it, although I have now the memory that it was quite popular in the olden days (at least, I seem to remember that it cropped up in every other conversation back when I was reading that book 20 or more years ago), and I had not heard anything about it for about that long.
So, the Mills number is that (or any of the) amazing real number A>1 with the property that
is a prime number for every positive integer n.
As one can expect, the doubly-exponential growth means that it would be pointless to try to use this to produce prime numbers. And one may guess that the proof of the existence of such a number has little to do with primes, and should apply to many other sequences of positive integers.
This is indeed so, but not in a completely trivial manner. More precisely, what the proof shows is that, given an infinite subset S of integers, and a real number c>1, one can find B, depending of course on the set and on c, such that
provided the set S has the property that, for some real number
and all large enough x, the intersection
is not empty. In other words, since θ<1, there must be some element of the set in all “short” intervals (from some point on), where “short” has the usual meaning in analytic number theory: the length is a power less than 1 of the left-hand extremity.
(Note that the relation between c and θ shows that, if we know a suitable value of θ for S, then we can always find a value of c that works, always assuming θ<1.)
What about primes, then? Do primes exist in short intervals? The answer is, indeed, yes, and it has been known to be so since the work of Hoheisel in 1930, but this is by no means a triviality! Indeed, if one looks at the problem from too far away, analyzing the number of primes in such an interval with the “explicit formulas” in terms of zeros of the Riemann zeta function, then one gets the impression that one will prove
(which is the expected answer, because of the Prime Number Theorem) only for θ>1/2, and only by knowing that
which we know only for θ=1. This means that, from the point of view of immediate consequences of the location of zeros of the zeta function, having primes in short intervals is comparable with having a zero-free strip.
From this point of view, we see that the existence of the Mills number is quite an interesting fact. Moreover, the smaller the value of c one can take, the shorter the intervals we manage to find primes in. The value c=3 which I quoted at the beginning is possible because the current best result about primes in short intervals states that, for x large enough,
contains the “right number” of primes. (In fact, this allows any c>12/5). This result is due to Huxley, and hasn’t been improved since 1972; however, if one wants only the existence of a positive proportion among the right number of primes, Baker and Harman have the record value 0.534 (this was in 1996, and allows c>2.14…).
All the proofs since Hoheisel’s time depend crucially on a way to get around the Riemann Hypothesis known as “density theorems” for the zeros. This is a fairly inconvenient name, since “density” might suggest “lots and lots of zeros everywhere”, whereas the intent and purpose of density theorems is to show that, although there might be zeros off the critical line, or even close to 1 (which is were they would fight against the pole of the Riemann zeta function, which is the White Knight that tries to produce primes, glorious primes), there can not be too many. The precise argument is presented in Chapter 10 of my book with H. Iwaniec. Note that density theorems have many other applications: certain particularly subtle ones for Dirichlet characters (“log-free density theorems”), the first of which was proved by Linnik, are crucial to the known proofs of his marvelous theorem according to which, for some absolute constant C>0, the smallest prime P(q,a) congruent to a modulo q, for a coprime with q, satisfies
(The best result here allows you to take any C>5.5 for q large enough, due to Heath-Brown; the Generalized Riemann Hypothesis gives this for any C>2). If this remains too mundane — some people do not like primes in arithmetic progressions –, note that you need similar theorems for cusp forms to give an upper bound of the right order of magnitude for the rank of the Jacobian J0(q) of the modular curve X0(q) for q prime, a result of P. Michel and myself.
Now for the proof of the existence of the Mills number, in the generality of a set S containing elements in short intervals. I won’t give all details, but here’s a sketch:
(1) define b(1) to be the smallest element of S above the point after which all short intervals contain at least one element of S;
(2) define inductively b(n+1) to be such that
(3) show, using the condition
that if we define
we then have
and deduce that the limit B of xn exists, and gives the desired general Mills number…