Prime gaps

If [math]\displaystyle{ p_n }[/math] denotes the n^th prime, then [math]\displaystyle{ p_{n+1}-p_n }[/math] is the n^th prime gap.

On average, the prime number theorem tells us that [math]\displaystyle{ p_{n+1}-p_n }[/math] has size [math]\displaystyle{ O(\log p_n) }[/math].

A recent result of Goldston-Pintz-Yildirim shows that there exist infinitely many n for which the gap is as small as [math]\displaystyle{ o(\log p_n) }[/math] (in fact more precise bounds are known). But the set of small gaps established by this method is sparse.

Cramer's conjecture asserts that the prime gap never exceeds [math]\displaystyle{ O(\log^2 p_n) }[/math] in size. If so, this resolves the finding primes project positively. However, the best upper bound on the prime gap is [math]\displaystyle{ O( p_n^{1/2} \log p_n ) }[/math] assuming the Riemann hypothesis, and [math]\displaystyle{ O( p_n^{0.525} ) }[/math] otherwise (a result of Baker, Harman, and Pintz; an earlier bound of [math]\displaystyle{ O(p_n^{0.535}) }[/math] was obtained by Baker and Harman.).

Rankin showed that the prime gap can be as large as [math]\displaystyle{ \log p_n \frac{\log \log p_n \log \log \log \log p_n}{(\log \log \log p_n)^3} }[/math].

1. R. C. Baker and G. Harman, “The difference between consecutive primes,” Proc. Lond. Math. Soc., series 3, 72 (1996) 261–280. MR 96k:11111
2. K. Soundararajan, Small gaps between prime numbers: The work of Goldston-Pintz-Yildirim
3. The Wikipedia entry on prime gaps