On the arithmetic average of the first $n$ primes

The arithmetic average of the first $n$ primes, $\bar p_n = {1\over n} \sum_{i=1}^n p_i$, exhibits very many \break 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 $p_n \longleftrightarrow \bar p_n$ must ultimately carry exactly the same information. But the averaged sequence $\bar p_n$, while very closely correlated with the primes, ($\bar p_n \sim {1\over2} p_n$), 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.