ON HÖLDER MAPS AND PRIME GAPS

Abstract

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 Holder continuity of this function is equivalent to a parameterised family of Cramer type estimates on the gaps between successive primes. Here the parameterisation comes from the Holder exponent. In particular, we show that Cramer'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 Holder of all orders but not Lipshitz and this is independent of Cramer's conjecture.