On the arithmetic average of the first $n$ primes

TL;DR

This paper studies the properties of the average of the first n primes, showing that many prime conjectures become theorems when applied to this smoothed sequence, highlighting the smoothing effect of averaging.

Contribution

The paper proves that prime-averaged analogues of several famous conjectures are actually theorems, demonstrating the smoothing effect of averaging on prime-related properties.

Findings

Prime-averaged analogues of conjectures are theorems.

Averaging primes produces a smoother sequence with better-behaved properties.

Local fluctuations differ significantly from the original prime sequence.

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.