On the geometric mean of the first n primes

TL;DR

This paper proves that the ratio of the nth prime to the geometric mean of the first n primes approaches Euler's number e, providing a precise asymptotic expansion and bounds for the error term.

Contribution

It offers a short proof of the limit of the ratio p_n/s_n approaching e and derives an explicit asymptotic formula with bounds for the correction term.

Findings

p_n/s_n converges to e as n increases

Derived an asymptotic expansion for p_n/s_n involving 1/log p_n

Provided explicit bounds for the error term in the expansion

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.