On the arithmetic average of the first $n$ primes

The arithmetic average of the first $n$ primes, ${\bar{p}}_n=\frac{1}{n}{\sum }_{i=1}^n{p}_i$, exhibits very many interesting and subtle properties. Since the transformation from $p_n\to {\bar{p}}_n$ is extremely easy to invert, $p_n=n{\bar{p}}_n-(n-1){\bar{p}}_{n-1}$, it is clear that these two sequences $p_n⟷{\bar{p}}_n$ must ultimately carry exactly the same information. But the averaged sequence ${\bar{p}}_n$, while very closely correlated with the primes, (${\bar{p}}_n\sim \frac{1}{2}{p}_n$), is much "smoother’‘, and much better behaved. Using extensions of various standard results I shall demonstrate that the prime-averaged sequence ${\bar{p}}_n$ satisfies prime-averaged analogues of the Cramer, Andrica, Legendre, Oppermann, Brocard, Fourges, Firoozbakht, Nicholson, and Farhadian conjectures. (So these prime-averaged analogues are not conjectures, they are theorems.) The crucial key to enabling this pleasant behaviour is the "smoothing’’ process inherent in averaging. Whereas the asymptotic behaviour of the two sequences is very closely correlated the local fluctuations are quite different.