Azar y Aritmetica

TL;DR

This paper introduces probabilistic number theory by studying the distribution and expected value of the number of prime divisors of a random integer, using accessible methods without advanced prerequisites.

Contribution

It provides an accessible introduction to probabilistic number theory, including foundational topics and sieve theory applications, without requiring prior advanced knowledge.

Findings

Analysis of the expected value of omega(n)

Distribution limits of omega(n) for large N

Probability of large deviations from the expected value

Abstract

Let omega(n) be the number of prime divisors of an integer n. Let n be an integer taken at random between 1 and N. What can be said about the value then taken by omega(n)? What is its expected value? What is its distribution in the limit? What is the probability that omega(n) will deviate greatly from its expected value? We will study these questions as an introduction to probabilistic number theory. We treat several central topics in probabilistic number theory without assuming previous knowledge of the area. Neither measure theory nor complex analysis are assumed. In the exercises, among other topics, we develop some of the bases of sieve theory as an application of probabilistic ideas. ----- Sea omega(n) el numero de divisores primos de un entero n. Sea n un entero tomado al azar entre 1 y N. Que se puede decir del valor que entonces tomara' omega(n)? Cual es su esperanza? Cual es su distribucion en el limite? Cual es la probabilidad que omega(n) tome valores que se alejen mucho de su esperanza? Estudiamos estas preguntas a guisa de introduccion a la teoria de numeros probabilistica. Trataremos varios topicos centrales de la teoria de probabilidades sin suponer conocimientos previos en el area. No asumiremos ni teoria de la medida ni analisis complejo. En los ejercicios, entre otros topicos, se desarrollaran las bases de la teoria de cribas como una aplicacion de ideas probabilisticas.