Carmichael number with three prime factors.

D R Heath-Brown (1)

1 MI - Mathematical Institute [Oxford]

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.

Mots clés: Ramanujan sum; Upper bound; Three prime factors; Carmichael numbers

Domaines: Mathématiques [math]

D R Heath-Brown. Carmichael number with three prime factors.. Hardy-Ramanujan Journal, 2007, Volume 30 - 2007, pp.6-12. ⟨10.46298/hrj.2007.156⟩. ⟨hal-01112050⟩