What is the probability there is no prime between $n$ and $n+\ln(n)$?

Question

Consequences of the Prime Number Theorem tell us the probability of $n$ being prime is $1/\ln(n)$. This also means that the number of expected primes between $n$ and $n+\ln(n)$ is close to $1$, but not always. What is the probability there is no prime between $n$ and $n+\ln(n)$?

Answer 1

This is currently a question we cannot answer unconditionally; however, there is a heuristic answer, which we can prove if we assume another result.

The naive probabilistic model is that each integer $k$ between $n$ and $n+\log n$ independently has a probability $1/\log k \sim 1/\log n$ of being prime. According to this model, the probability that none of these integers are prime (as alluded to in Sil's comment) is $$ \prod_{n<k<n+\log n} \bigg( 1-\frac1{\log n} \bigg) \sim \bigg( 1-\frac1{\log n} \bigg)^{\log n} \sim e^{-1}. $$ But this model actually gives something more: for any nonnegative integer $m$, the probability that there are exactly $m$ primes between $n$ and $n+\log n$ is $e^{-1}/m!$. In other words, intervals of this length should give rise to a Poisson distribution. (And this can be generalized to the primes between $n$ and $n+C\log n$ for any constant $C>0$.)

In 1976, Gallagher proved that this is indeed the case if you assume a suitably strong version of the Hardy–Littlewood prime $k$-tuples conjecture. This paper of Goldston and Ledoan describes the result more precisely on its first page and gives the exact reference.

One can criticize the naive model by noting that adjacent integers can never both be prime, nor can three consecutive integers, etc. But a more advanced probabilistic model that takes divisibility by small primes into account eventually leads to the same heuristic answer. And Gallagher's result shows that the naive heuristic is correct on average on this scale (I doubt it would be on significantly smaller scales, for instance).