Variants on Andrica's conjecture with and without the Riemann hypothesis

TL;DR

This paper explores variants of Andrica's conjecture, examining the implications of the Riemann hypothesis and providing new bounds and unconditional results related to prime gaps and prime distributions.

Contribution

It introduces new bounds on prime gaps under the Riemann hypothesis and refines unconditional results, extending the understanding of Andrica's conjecture variants.

Findings

Under Riemann hypothesis, bounds involving roots of primes are established.

Unconditional results on differences of logarithmic powers of consecutive primes are provided.

The region where Andrica's conjecture is verified is slightly expanded.

Abstract

The gap between what we can explicitly prove regarding the distribution of primes and what we suspect regarding the distribution of primes is enormous. It is (reasonably) well-known that the Riemann hypothesis is not sufficient to prove Andrica's conjecture: $\forall n\geq 1$, is $\sqrt{p_{n+1}}-\sqrt{p_n} \leq 1$? But can one at least get tolerably close? I shall first show that with a logarithmic modification, provided one assumes the Riemann hypothesis, one has \[ {\sqrt{p_{n+1}}\over\ln p_{n+1}} -{\sqrt{p_n}\over\ln p_n} < {11\over25}; \qquad (n\geq1). \] Then, by considering more general $m^{th}$ roots, again assuming the Riemann hypothesis, I shall show that \[ {\sqrt[m]{p_{n+1}}} -{\sqrt[m]{p_n}} < {44\over25 \,e\, (m-2)}; \qquad (n\geq 3;\; m >2). \] In counterpoint, if we limit ourselves to what we can currently prove unconditionally, then the only explicit Andrica-like results seem to be variants on the results below: \[ \ln^2 p_{n+1} - \ln^2 p_n < 9; \qquad (n\geq1). \] \[ \ln^3 p_{n+1} - \ln^3 p_n < 52; \qquad (n\geq1). \] \[ \ln^4 p_{n+1} - \ln^4 p_n < 991; \qquad (n\geq1). \] I shall also slightly update the region on which Andrica's conjecture is unconditionally verified.