Average Prime Gap

The average of the prime gaps up to the th prime number is the arithmetic mean of the first prime gaps, so

(1)

Here, the bar denotes the average. The equality on the right follows because the sum is a telescoping sum. Each interior prime cancels, leaving . The prime number theorem, in the equivalent form , therefore gives

(2)

Thus the average gap grows without bound. Individual gaps fluctuate substantially around this scale. In particular, Goldston et al. (2009) proved that the limit inferior

(3)

and the stronger result on bounded gaps between primes gives infinitely many gaps no larger than 246 (Polymath 2014). Neither result says that the average prime gap is bounded.