On the arithmetic and geometric means of the prime numbers

TL;DR

This paper derives improved bounds for the ratio of the arithmetic to geometric means of prime numbers and proves related conjectures using explicit estimates of prime-related functions.

Contribution

It provides the first explicit bounds for the ratio of means of primes and confirms several conjectures by Hassani.

Findings

Established explicit upper and lower bounds for the ratio of prime means.

Proved several conjectures related to the ratio of prime means.

Utilized explicit estimates of prime counting functions and related functions.

Abstract

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.