  In this paper we establish explicit upper and lower bounds for the ratio of
the arithmetic and geometric means of the prime numbers, which improve the
current best estimates. Further, we prove several conjectures related to this
ration stated by Hassani. In order to do this, we use explicit estimates for
the prime counting function, Chebyshev's $\vartheta$-function and the sum of
the first $n$ prime numbers.
