Behaviour of the sequence ${\vartheta }_n=\vartheta (p_n)$

The well-known sequence ${\vartheta }_n=\vartheta (p_n)={\sum }_{i=1}^n\ln p_i=\ln ([p_n]\mathrm{\#})$ exhibits numerous extremely interesting properties. Since $p_n=\exp ({\vartheta }_n-{\vartheta }_{n-1})$, it is immediately clear that the two sequences $p_n⟷{\vartheta }_n$ must ultimately encode exactly the same information. But the sequence ${\vartheta }_n$, while being extremely closely correlated with the primes, (in fact, ${\vartheta }_n\sim p_n$), is very much better behaved than the primes themselves. Using numerous suitable extensions of various reasonably standard results, I shall demonstrate that the sequence ${\vartheta }_n$ satisfies suitably defined $\vartheta$-analogues of the usual Cramer, Andrica, Legendre, Oppermann, Brocard, Firoozbakht, Fourges, Nicholson, and Farhadian conjectures. (So these $\vartheta$-analogues are not conjectures, they are instead theorems.) The crucial key to enabling this pleasant behaviour is the regularity (and relative smallness) of the $\theta$-gaps ${\mathfrak{g}}_n={\vartheta }_{n+1}-{\vartheta }_n=\ln p_{n+1}$. While superficially these results bear close resemblance to some recently derived results for the averaged primes, ${\bar{p}}_n=\frac{1}{n}{\sum }_{i=1}^n{p}_i$, both the broad outline and the technical details of the arguments given and proofs presented are quite radically distinct.