Primos, paridad y análisis

Abstract: Distinguir entre enteros con un n\'umero par o impar de divisores primos es una de las tareas m\'as dif\'iciles en la teor\'ia anal\'itica de n\'umeros. Un trabajo reciente de Matom\"aki y Radziwi{\l}{\l} muestra que, en promedio, ambos existen con la misma frecuencia a\'un en intervalos muy cortos. Este avance ya ha tenido varias aplicaciones importantes en las manos de Matom\"aki, Radziwi{\l}{\l}, Tao y Ter\"av\"ainen. Explicaremos en detalle una prueba completa del resultado original de Matom\"aki y Radziwi{\l}{\l}, as\'i como de varias aplicaciones. ----- To distinguish between integers with an even or an odd number of prime factors is one of the most difficult tasks in Analytic Number Theory. A recent work by Matom\"aki and Radziwi{\l}{\l} shows that, in average, both types of integers appear with the same frequency even in very short intervals. This breakthrough has already had several applications in the hands of Matom\"aki, Radziwi{\l}{\l}, Tao and Ter\"av\"ainen. We explain in detail the complete proof of both the original result by Matom\"aki and Radziwi{\l}{\l} and of some of its applications.