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

TL;DR

The paper presents an elementary route to the asymptotic for the partial sums of the reciprocals of primes by exploiting the von Mangoldt function. It first establishes the key identity $\log(n!) = \sum_{m\le n} \Lambda(m)\left\lfloor \frac{n}{m} \right\rfloor$ (via Legendre) and the related bound $\sum_{m\le x} \frac{\Lambda(m)}{m} = \log x + O(1)$, then uses Abel summation to derive $\sum_{p\le x} \frac{1}{p} = \log\log x + b + O\left(\frac{1}{\log x}\right)$, with $b$ explicit in terms of constants and primes. An application to integers with large prime factors yields a density limit $\log 2$ for such numbers. The work emphasizes the complementarity of arithmetic functions and offers a pedagogical alternative to PNT-based proofs, while connecting to constants like Meissel-Mertens $M$ and suggesting extensions via Selberg-type identities.

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 $Λ(m)/m$, where $Λ$ denotes the von Mangoldt function, with an application of Abel's summation formula and properties of the second Chebyshev function $Ψ(x)=\sum_{n\le x}Λ(n)$. 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.