New Proof That the Sum of the Reciprocals of Primes Diverges

Vicente Jara-Vera,Carmen Sánchez-Ávila

doi:10.3390/math8091414

Abstract

In this paper, we give a new proof of the divergence of the sum of the reciprocals of primes using the number of distinct prime divisors of positive integer n, and the placement of lattice points on a hyperbola given by n=pr with prime number p. We also offer both a new expression of the average sum of the number of distinct prime divisors, and a new proof of its divergence, which is very intriguing by its elementary approach.

Highlights

It has sometimes been mentioned that number theory is the theory of prime numbers
There are infinitely many prime numbers, a theorem proved by Euclid, but it was proved by other different methods by Christian Goldbach, Leonhard Euler, Charles Hermite, and Thomas J
In terms of Π( x ), Euclid’s theorem on the infinitude of prime numbers can be expressed as lim Π( x ) = +∞

Summary

Introduction

It has sometimes been mentioned that number theory is the theory of prime numbers. There are infinitely many prime numbers, a theorem proved by Euclid (ca. 300 BCE), but it was proved by other different methods by Christian Goldbach, Leonhard Euler, Charles Hermite, and Thomas J. A main element of our proof is function ω (n), which is defined as the number of distinct prime divisors of n, which is additive but not completely additive. Rāmānujan [13] proved that, if the number of distinct prime divisors of n, ω (n) is additively calculated up to a certain value,. The latter is the result that we need for our proof of (1). From there, we analyze the of the interval, of value ω convergence or divergence of the average sum of the number of distinct prime divisors of n.

With our choice of variables x b c

If we consider that