Abstract

In this paper we will present an inequality with the sequence of prime numbers 2,3,5,7.... We prove that there exists a positive constant real number k, such that for every real number λ>k, there exists a natural number nλ, such that for every natural number n>nλ, it is true the inequality 2/2-1.3/3-1...Pn/Pn-1<λPn/n, where Pn is the n-th prime number. The constant number k is equal with eM+σ, where M is the Merten’s constant and σ is the sum of the convergent series ∑1/p(p-1). The constant k has an approximate value k≈2.812.

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References

[1] Tom M. Apostol, “An asymptotic formula for the partial sums Σp≤x(1/p)”, Introduction to Analytic Number Theory, pp 88, Springer-Verlag, New York Heidelberg Berlin 1976.
[2] A. Languasco, A. Zaccagnini, A note of Merten’s formula for arithmetic progressions, Jurnal of Number Theory, Volume 127, Issue 1, November 2007, pp 37-46.
[3] Tom M. Apostol, “Some elementary theorems on the distribution of prime numbers”, Introduction to Analytic Number Theory, pp 74, Springer-Verlag, New York Heidelberg Berlin 1976.