D R Heath-Brown - Carmichael number with three prime factors.

hrj:156 - Hardy-Ramanujan Journal, January 1, 2007, Volume 30 - https://doi.org/10.46298/hrj.2007.156
Carmichael number with three prime factors.

Authors: D R Heath-Brown

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.

Volume: Volume 30
Published on: January 1, 2007
Submitted on: March 3, 2015
Keywords: Upper bound, Three prime factors,Carmichael numbers, Ramanujan sum,[MATH] Mathematics [math]


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