Do Arithmetic Mean and Geometric Mean of Prime Numbers converge?

I was looking at a list of primes. I noticed that AM(p1,p2,,pn)pn seemed to converge.

This led me to try GM(p1,p2,,pn)pn which also seemed to converge.

I did a quick Excel graph and regression and found the former seemed to converge to 12 and latter to 1e. 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 12).

Are these observations correct and are there any proofs towards:

lim


{
\lim_{n\to\infty} \left( \frac{GM (p_1, p_2, \ldots, p_n)}{p_n} \right)
= \frac{1}{e} \tag2
}

Also, does the limit
{
\lim_{n\to\infty} \left( \frac{HM (p_1, p_2, \ldots, p_n)}{p_n} \right) \tag3
}
exist?

Answer

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 kth 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.,
p_n \approx n \log n \quad \textrm{as} \ n \to \infty
as well as an inequality with Chebyshev’s function \theta(x) = \sum_{p \le x}\log p
where p are primes less than x.

Attribution
Source : Link , Question Author : Soham , Answer Author : I. J. Kennedy

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