On the geometric mean of the first n primes

Abstract

Let $p_n$ be the $n$th prime, and consider the sequence $s_n = (2\cdot3\cdots p_n)^{1/n} = (p_n\#)^{1/n}$, the geometric mean of the first $n$ primes. We give a short proof that $p_n/s_n \to e$, a result conjectured by Vrba (2010) and proved by Sandor and Verroken (2011). We show that $p_n/s_n = \exp(1+1/\log p_n + O(1/\log^2 p_n))$ as $n\to\infty$, and give explicit lower and upper bounds for the $O(1/\log^2 p_n)$ term.