On the geometric mean of the first n primes

Abstract

Let $p_n$ be the $n$th prime, and consider the sequence $s_n=(2\cdot 3\cdots p_n{)}^{1/n}=(p_n\#{)}^{1/n}$, 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.

4 pages, 1 table