On the arithmetic average of the first $n$ primes

Abstract

The arithmetic average of the first $n$ primes, $\bar p_n = {1\over n} \sum_{i=1}^n p_i$, 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 $p_n \longleftrightarrow \bar p_n$ 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.