I have problems in understanding the following assertion:
We introduce a probability distribution on the natural numbers $\mathbb{N} - {0}$ as follows: $$P(I) = c_s\sum_{k\in I} \frac{1}{|k|^{1+2s}}$$
Where $I\subseteq N$
I don't know if I got the meaning well but is this the probability to pick a random natural number?
Also what is the $s$ term in the exponent?
The distribution you have is known as Zipf's distribution, formally speaking, for $s>1$ we have $$P(I\subseteq \mathbb{N})=c_s\sum_{k\in I}\frac{1}{k^{s+1}}$$ where $$c_s=\frac{1}{\displaystyle \sum_{k=1}^\infty\frac{1}{k^{s+1}}}$$ reason by wich it is also known as Zeta or Riemann's distribution, since $c_s$ is expressed as a fraction of the Riemann's function. So the necesity of taking $s>1$ is due to the convergence of the Riemann's function.
Applications of this distribution can be enumerated in ranking areas as linguistic(Zipf), family incomes in a given country(Pareto) and many others. You can read about it in
Univariate Discrete Distributionsof Samuel Kotz, from Wiley Series in Probability and Statistics.Also i advice you to take a look at Zipf's Law, although it is a little harder.