On H\"older maps and prime gaps

TL;DR

This paper links the regularity of a function mapping reciprocals of integers to reciprocals of primes with prime gap conjectures, showing that certain continuity properties are equivalent to famous conjectures like Cramér's.

Contribution

It establishes a novel equivalence between H"older continuity of a prime-related function and Cramér-type prime gap estimates, providing a new perspective on prime distribution conjectures.

Findings

H"older continuity of the function is equivalent to Cramér's conjecture.

The inverse map is H"older of all orders but not Lipschitz, independent of Cramér's conjecture.

Cramér's conjecture is equivalent to the map being Lipschitz.

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 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.