Variants on Andrica's conjecture with and without the Riemann hypothesis
Matt Visser
TL;DR
This paper investigates near-Sharp variants of Andrica’s conjecture under the Riemann hypothesis and unconditionally. By leveraging the explicit prime-gap bound $g_n < {22\over25}\sqrt{p_n}\ln p_n$ from Carneiro–Milinovich–Soundararajan, it proves a logarithmic modification of Andrica and a higher-root generalization, yielding concrete, RH-assisted bounds on gaps of $\sqrt[m]{p_n}$. It then derives fully explicit unconditional bounds on differences of powers of $\ln p_n$, and demonstrates unconditional numerical verification of Andrica and the CMS inequality up to just before the 81st maximal prime gap, i.e., for primes below about $1.836\times10^{19}$. The results illustrate how RH can push Andrica-type bounds closer to the conjectured threshold, while outlining the challenges of obtaining unconditional proofs and the value of explicit numerical checks at extreme scales.
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.