# Do Arithmetic Mean and Geometric Mean of Prime Numbers converge?

I was looking at a list of primes. I noticed that $\frac{AM (p_1, p_2, \ldots, p_n)}{p_n}$ seemed to converge.

This led me to try $\frac{GM (p_1, p_2, \ldots, p_n)}{p_n}$ which also seemed to converge.

I did a quick Excel graph and regression and found the former seemed to converge to $\frac{1}{2}$ and latter to $\frac{1}{e}$. As with anything related to primes, no easy reasoning seemed to point to those results (however, for all natural numbers it was trivial to show that the former asymptotically tended to $\frac{1}{2}$).

Are these observations correct and are there any proofs towards:

Also, does the limit exist?

Your conjecture for GM was proved in 2011 in the short paper On a limit involving the product of prime numbers by József Sándor and Antoine Verroken.

Abstract. Let $p_k$ denote the $k$th prime number. The aim of this note is to prove that the limit of the sequence $(p_n / \sqrt[n]{p_1 \cdots p_n})$ is $e$.

The authors obtain the result based on the prime number theorem, i.e.,

as well as an inequality with Chebyshev’s function
where $p$ are primes less than $x$.