New Proof That the Sum of the Reciprocals of Primes Diverges

Vicente Jara-Vera and Carmen Sánchez-Ávila
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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

Mathematics, 2020, vol. 8, issue 9, 1-9

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 = p r 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.

Keywords: number theory; primes; reciprocals of primes (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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