On Hölder maps and prime gaps

TL;DR

The paper investigates how Hölder regularity of the reciprocal-prime map $f(n)=1/p_n$ (and its inverse) encodes prime-gap information. By combining Rosser-type bounds, Jensen-type inequalities, and known prime-gap results, it establishes precise equivalences between Hölder continuity properties of these maps and parametrised Cramér-type bounds on $p_{n+1}-p_n$, linking metric-geometry notions to classic conjectures in number theory. Key contributions include: (i) the inverse map $1/p_n\mapsto 1/n$ is Hölder of all orders but not Lipschitz; (ii) the forward map $1/n\mapsto 1/p_n$ is Lipschitz on large scales and its full Hölder regularity corresponds to a family of prime-gap bounds; (iii) Cramér's conjecture is equivalent to the forward map being Lipschitz, with explicit corollaries derived from Baker–Harman–Pintz giving concrete exponents in large ranges. These results provide a novel geometric-analytic lens on prime gaps, offering a framework to interpret and test Cramér-type bounds via Hölder regularity.

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

Theorems & Definitions (13)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.1
• Theorem 1.3
• Theorem 1.4
• Corollary 1.2
• proof
• proof : Proof of Theorem \ref{['LOWER.BOUND.']}
• Proposition 3.1
• proof