On H\"older maps and prime gaps

Let $p_n$ denote the $n$th prime, and consider the function $1/n \mapsto 1/p_n$ which maps the reciprocals of the positive integers bijectively to the reciprocals of the primes. We show that H\"older continuity of this function is equivalent to a parameterised family of Cram\'er type estimates on the gaps between successive primes. Here the parameterisation comes from the H\"older exponent. In particular, we show that Cram\'er's conjecture is equivalent to the map $1/n \mapsto 1/p_n$ being Lipschitz. On the other hand, we show that the inverse map $1/p_n \mapsto 1/n$ is H\"older of all orders but not Lipshitz and this is independent of Cram\'er's conjecture.