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

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.