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On the Arithmetic and Geometric Means of the First n Prime Numbers

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Abstract

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

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  1. Institute of Mathematics, Heinrich-Heine University Düsseldorf, 40225, Düsseldorf, Germany

    Christian Axler

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  1. Christian Axler
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Correspondence to Christian Axler.

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Axler, C. On the Arithmetic and Geometric Means of the First n Prime Numbers. Mediterr. J. Math. 15, 93 (2018). https://doi.org/10.1007/s00009-018-1137-5

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  • DOI: https://doi.org/10.1007/s00009-018-1137-5

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