Probabilidades y distribución de los números primos

Abstract

In this project, we will try to study the prime factors of a number taken at random. The main objective will be to introduce the probabilistic theory of numbers and, at the same time, several central themes of probability theory in general: expected value, variance and the central limit. The story of probabilistic number theory begins with Hardy and Ramanujan, who were the ones who pushed the study of the central theme of this project: the distribution of ω(n), the number of prime divisors of a natural number n. Later generations, the area continued to develop remarkably with Erdös and Kac’s theorem (which is the main theme we will try to demonstrate and understand) until this day, thanks to both number theory specialists and probability experts. We will not assume any knowledge of complex analysis or measure theory. We only need to consider finite spaces of probability, so that we will start by studying expected value by treating ω as a random variable, using Chebyshev’s inequality. Subsequently we will study the variance, and aply it in two ways: one using the inequality of Chebyshev, and another somewhat more useful (that is, it will give better theoretical results). Finally we will apply the central limit theorem, in order to try to understand and demonstrate the aforementioned Erdös - Kac theorem.