{"docId":156,"paperId":156,"url":"https:\/\/hrj.episciences.org\/156","doi":"10.46298\/hrj.2007.156","journalName":"Hardy-Ramanujan Journal","issn":"","eissn":"2804-7370","volume":[{"vid":41,"name":"Volume 30 - 2007"}],"section":[],"repositoryName":"Hal","repositoryIdentifier":"hal-01112050","repositoryVersion":1,"repositoryLink":"https:\/\/hal.archives-ouvertes.fr\/hal-01112050v1","dateSubmitted":"2015-03-03 16:13:58","dateAccepted":"2015-06-12 16:05:52","datePublished":"2007-01-01 08:00:00","titles":{"en":"Carmichael number with three prime factors."},"authors":["Heath-Brown , D R"],"abstracts":{"en":"Let $C_3(x)$ be the number of Carmichael numbers $n\\le x$ having exactly 3 prime factors. It has been conjectured that $C_3(x)$ is of order $x^{1\/3}(\\log x)^{-1\/3}$ exactly. We prove an upper bound of order $x^{7\/20+\\varepsilon}$, improving the previous best result due to Balasubramanian and Nagaraj, in which the exponent $7\/20$ was replaced by $5\/14$. The proof combines various elementary estimates with an argument using Kloosterman fractions, which ultimately relies on a bound for the Ramanujan sum."},"keywords":[["Upper bound"],["Three prime factors"],["Carmichael numbers"],["Ramanujan sum"],"[MATH] Mathematics [math]"]}