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New Proof That the Sum of the Reciprocals of Primes Diverges

Author

Listed:
  • 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)

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.

Suggested Citation

  • Vicente Jara-Vera & Carmen Sánchez-Ávila, 2020. "New Proof That the Sum of the Reciprocals of Primes Diverges," Mathematics, MDPI, vol. 8(9), pages 1-9, August.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:9:p:1414-:d:403023
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