Carmichael number with three prime factors.

Abstract

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.

Locations

• HAL (Le Centre pour la Communication Scientifique Directe) - View - PDF
• Hardy-Ramanujan Journal - View - PDF