Asymptotic equivalents of partial sums of the reciprocals of prime numbers via the von Mangoldt function

TL;DR

This paper presents a new elementary proof for the asymptotic behavior of the sum of reciprocals of primes, using the von Mangoldt function and Abel's summation, offering insights into prime number theory.

Contribution

It introduces an alternative, elementary approach to derive the asymptotic equivalent of the sum of reciprocals of primes, complementing classical methods.

Findings

Derived an asymptotic equivalent for the sum of reciprocals of primes.

Connected the result to properties of the von Mangoldt function and Chebyshev functions.

Provided a pedagogical perspective on elementary methods in number theory.

Abstract

In this paper, we discuss an alternative approach to determine an asymptotic equivalent of the partial sum of the reciprocals of prime numbers. This well-known result, related to Merten's second theorem, is usually derived through methods similar to those found in Hardy and Wright's book ``An introduction to the theory of numbers'', involving comparisons with integrals. The present proof differs in several respects, combining an equivalent for the partial sum of $\Lambda(m)/m$, where $\Lambda$ denotes the von Mangoldt function, with an application of Abel's summation formula and properties of the second Chebyshev function $\Psi(x)=\sum_{n\le x}\Lambda(n)$. A simple application to the study of integers with large prime factors is also presented. Beyond the pedagogical aspect of this work, the aim is to highlight the complementarity of arithmetic functions and to show that interesting (and nontrivial) results can be obtained by means of elementary methods.