Mathematics > Number Theory
[Submitted on 7 Apr 2018 (v1), last revised 28 Nov 2018 (this version, v3)]
Title:Variants on Andrica's conjecture with and without the Riemann hypothesis
View PDFAbstract: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.
Submission history
From: Matt Visser [view email][v1] Sat, 7 Apr 2018 03:26:57 UTC (11 KB)
[v2] Tue, 18 Sep 2018 22:05:49 UTC (11 KB)
[v3] Wed, 28 Nov 2018 20:06:23 UTC (12 KB)
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