On the arithmetic and geometric means of the prime numbers

Christian Axler

On the arithmetic and geometric means of the prime numbers

Number Theory 2017-09-05 v2

Authors: Christian Axler

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.

Keywords

prime numbers arithmetic mean geometric mean prime counting function Chebyshev's theta function explicit bounds

Cite

@article{arxiv.1609.07934,
  title  = {On the arithmetic and geometric means of the prime numbers},
  author = {Christian Axler},
  journal= {arXiv preprint arXiv:1609.07934},
  year   = {2017}
}

Comments

v2: 14 pages; better results

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