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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.
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https://hal.archives-ouvertes.fr/hal-01112050
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Submitted on : Monday, February 2, 2015 - 11:08:51 AM
Last modification on : Monday, March 28, 2022 - 8:14:08 AM
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D R Heath-Brown. Carmichael number with three prime factors.. Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 2007, Volume 30 - 2007, pp.6-12. ⟨10.46298/hrj.2007.156⟩. ⟨hal-01112050⟩

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