  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.
