Positive lower density of large prime gaps

The explanation

Look at the primes near 113. The next prime after 113 is 127, so the gap between them is 14. Near 113, primes are usually about 4.7 apart. So this gap is about three times the usual size. Such gaps are not rare. Among the first million primes, 31,946 gaps are more than three times the usual size. That is about 3.2%. For five times the usual size, the count is 2,420, about 0.24%. The share falls fast as the bar rises. The question is whether it keeps falling toward zero as the primes go on.

The usual size has a precise meaning. The prime number theorem, a classical result on how primes thin out, sets the average gap near a number p. That average is about log p, the natural logarithm of p. The natural log of 113 is about 4.7. The millionth prime is 15,485,863, and its log is about 16.6. So primes slowly spread out, and the yardstick grows with them.

Mathematicians knew long ago that single gaps can be huge. Westzynthius proved in 1931 that gaps can exceed any fixed multiple of log p. Erdős and Rankin then gave stronger and stronger lower bounds for the largest gaps. In 2016, Ford, Green, Konyagin and Tao, and separately Maynard, pushed these bounds further. The five of them then wrote a joint paper in 2018. But all of this concerns a few record gaps. A record gap says nothing about how often large gaps occur. The closest earlier result on frequency came from Bazzanella, Languasco and Zaccagnini in 2010. They got a positive share of gaps only for bars below about 0.579 times log p. That is smaller than an average gap.

The main obstacle is a counting trap. It is easier to count empty stretches of numbers than to count gaps. But one long gap contains many short empty stretchesSource 1. So a few giant gaps could produce many empty stretches. Counting stretches cannot tell many large gaps from a handful of enormous ones. Terence Tao pointed out this missing control in a 2019 online answer.

The paper avoids the trap with one extra demand. It looks at two windows of equal length placed side by side. The first window must contain a prime, and the second must contain none. Then the last prime in the first window starts a gap longer than one windowSource 2. Each prime can be picked this way only a limited number of times, however long its gap is. So many good window pairs force many distinct large gaps. The hard part is showing that such pairs are common. The paper builds special weights that keep the first window likely to hold a primeSource 3. At the same time, they make a prime in the second window very unlikely.

The main theorem says this. Fix any bar, say C times log p, for any positive number C. Then a fixed positive share of the first N gaps clears that barSource 4. This holds for every large enough N, not just for some lucky NSource 5. Mathematicians call this a positive lower density. The share may depend on CSource 5. In the example, the 0.24% at five times the yardstick cannot drift down to zero. The same holds at a hundred times the yardstick. The theorem does not say what the share actually is. The counts above are only data, not the theorem's constant.

There is also an old puzzle behind the result. Divide each prime by its position in the list. The first five are 2/1 = 2, 3/2 = 1.5, 5/3 ≈ 1.67, 7/4 = 1.75 and 11/5 = 2.2. These ratios creep upward on average, but they zigzag. A ratio rises at a step exactly when the next gap is larger than the current ratioSource 6. That ratio is roughly log p. So the theorem with a bar of twice the log handles itSource 7. In 1962, Erdős and Prachar asked whether the rising steps have positive lower density. Among the first million steps, 406,140 rise, about 41%. The paper answers their question with yesSource 8.

How sure is this? A computer proof checker called Lean has checked a formal version of the answer to Erdős and Prachar. It states that the steps where the ratio rises have positive lower density. The Lean scope note does not describe a separate check of the main theorem for every C. So for that general statement, the paper claims to prove it. No specialists have published a review yet.

What it doesn't show. This does not give the size of the share, and it does not find new record-size gaps.