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.
