{"docId":7433,"paperId":7433,"url":"https:\/\/hrj.episciences.org\/7433","doi":"10.46298\/hrj.2021.7433","journalName":"Hardy-Ramanujan Journal","issn":"","eissn":"2804-7370","volume":[{"vid":438,"name":"Volume 43 - Special Commemorative volume in honour of Srinivasa Ramanujan - 2020"}],"section":[],"repositoryName":"Hal","repositoryIdentifier":"hal-03208199","repositoryVersion":1,"repositoryLink":"https:\/\/hal.archives-ouvertes.fr\/hal-03208199v1","dateSubmitted":"2021-04-30 14:31:46","dateAccepted":"2021-05-06 14:29:16","datePublished":"2021-05-06 14:29:56","titles":{"en":"A localized Erd\u0151s-Kac theorem"},"authors":["Dixit, Anup B","Ram Murty, M"],"abstracts":{"en":"Let \u03c9_y (n) be the number of distinct prime divisors of n not exceeding y. If y_n is an increasing function of n such that log y_n = o(log n), we study the distribution of \u03c9_{y_n} (n) and establish an analog of the Erd\u0151s-Kac theorem for this function. En route, we also prove a variant central limit theorem for random variables, which are not necessarily independent, but are well approximated by independent random variables."},"keywords":[{"en":"prime divisors"},{"en":"Erd\u0151s-Kac theorem"},{"en":"central limit theorem"},"2010 Mathematics Subject Classification. 11N25, 11N64, 11K65, 60F05","[MATH]Mathematics [math]"]}