New Proof That the Sum of the Reciprocals of Primes Diverges
Vicente Jara-Vera,
Carmen Sánchez-Ávila
Affiliations
Vicente Jara-Vera
Department of Applied Mathematics to Information Technology and Communications (Telecommunication Engineering), Polytechnical University of Madrid, Avenida Complutense 30, 28040 Madrid, Spain
Carmen Sánchez-Ávila
Department of Applied Mathematics to Information Technology and Communications (Telecommunication Engineering), Polytechnical University of Madrid, Avenida Complutense 30, 28040 Madrid, Spain
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.
