On a limit involving the product of prime numbers

József Sándor and Antoine Verroken
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 17, 2011, Number 2, Page 1—3
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Authors and affiliations

József Sándor
Babeș-Bolyai University of Cluj, Romania

Antoine Verroken
Univ. of Gent, Gent, Belgium

Abstract

Let p_k denote the k-th prime number. The aim of this note is to prove that the limit of the sequence (p_n / \sqrt[n]{p_1 \cdots p_n}) is e.

Keywords

  • Arithmetic functions
  • Estimates
  • Primes

AMS Classification

  • 11A25
  • 11N37

References

  1. P. Dusart, The kth prime is greater than k(ln k + ln ln k − 1) for k ≥ 2; Math. Comp., 68(1999), no. 225, 411-415.
  2. J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math., 6(1962), 64-94.
  3. J. Sándor, On a limit for the sequence of primes, Octogon Math. Mag., 8(2000), no. 1, 180-181.
  4. J. Sándor, Geometric theorems, diophantine equations, and arithmetic functions, American Research Press, 2002, USA.

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Cite this paper

Sándor, J., & Verroken, A. (2011). On a limit involving the product of prime numbers. Notes on Number Theory and Discrete Mathematics, 17(2), 1-3.

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