Primos, paridad y an\'alisis

TL;DR

This paper discusses a recent breakthrough in analytic number theory showing that integers with an even or odd number of prime factors occur with equal frequency in short intervals, including detailed proofs and applications.

Contribution

It provides a complete proof of Matomäki and Radziwiłł's result and explores its significant applications in number theory.

Findings

Integers with even or odd prime factors appear equally often in short intervals.

The result has multiple important applications in analytic number theory.

The paper offers detailed proofs of the original theorem and its applications.

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.