The Divergence of the Prime Harmonic Series

 

Title: The Divergence of the Prime Harmonic Series
Author: Manuel Eberl
Submission date: 2015-12-28
Abstract:

In this work, we prove the lower bound ln(H_n) - ln(5/3) for the partial sum of the Prime Harmonic series and, based on this, the divergence of the Prime Harmonic Series ∑[p prime] · 1/p.

The proof relies on the unique squarefree decomposition of natural numbers. This is similar to Euler's original proof (which was highly informal and morally questionable). Its advantage over proofs by contradiction, like the famous one by Paul Erdős, is that it provides a relatively good lower bound for the partial sums.

BibTeX:
@article{Prime_Harmonic_Series-AFP,
  author  = {Manuel Eberl},
  title   = {The Divergence of the Prime Harmonic Series},
  journal = {Archive of Formal Proofs},
  month   = dec,
  year    = 2015,
  note    = {\url{http://isa-afp.org/entries/Prime_Harmonic_Series.shtml},
            Formal proof development},
  ISSN    = {2150-914x},
}
License: BSD License