2025/07/18 by Visser, Matt
#FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2507.14410
The well-known sequence ϑn = ϑ(pn) = ∑i=1n ln pi= ln([pn]#) exhibits numerous extremely interesting properties. Since pn = exp(ϑn - ϑn-1), it is immediately clear that the two sequences pn \longleftrightarrow ϑn must ultimately encode exactly the same information. But the sequence ϑn, while being extremely closely correlated with the primes, (in fact, ϑn ∼ pn), is very much better behaved than the primes themselves. Using numerous suitable extensions of various reasonably standard results, I shall demonstrate that the sequence ϑn satisfies suitably defined ϑ-analogues of the usual Cramer, Andrica, Legendre, Oppermann, Brocard, Firoozbakht, Fourges, Nicholson, and Farhadian conjectures. (So these ϑ-analogues are not conjectures, they are instead theorems.) The crucial key to enabling this pleasant behaviour is the regularity (and relative smallness) of the θ-gaps \mathfrakgn = ϑn+1-ϑn= ln pn+1. While superficially these results bear close resemblance to some recently derived results for the averaged primes, pn = 1\over n ∑i=1n pi, both the broad outline and the technical details of the arguments given and proofs presented are quite radically distinct.