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.