Mathematics – Probability
Scientific paper
2009-09-04
Mathematics
Probability
72 pages, based on course given at the Universidad Nacional Mayor de San Marcos (Lima, Peru); in Spanish; second version - min
Scientific paper
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.
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