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.
Categories: math.NT