Journal of Integer Sequences, Vol. 15 (2012), Article 12.3.2

On the Truncated Kernel Function


Jean-Marie De Koninck
Département de Mathématiques et de Statistique
Université Laval
Québec G1V 0A6
Canada

Ismaïla Diouf
Département de Mathématiques et d'Informatique
FST - Université Cheikh Anta DIOP
BP 5005, Dakar-Fann
Senegal

Nicolas Doyon
Département de Mathématiques et de Statistique
Université Laval
Québec G1V 0A6
Canada

Abstract:

We study properties of the truncated kernel function $ \gamma_2$ defined on integers $ n\ge 2$ by $ \gamma_2(n)=\gamma(n)/P(n)$, where $ \gamma(n)=\prod_{p\vert n}p$ is the well-known kernel function and $ P(n)$ is the largest prime factor of $ n$. In particular, we show that the maximal order of $ \gamma_2(n)$ for $ n\le x$ is $ (1+o(1))x/\log x$ as $ x\to \infty$ and that $ \sum_{n\le x} 1/\gamma_2(n)= (1+o(1)) \eta x/\log x$, where $ \eta=\zeta(2)\zeta(3)/\zeta(6)$. We further show that, given any positive real number $ u<1$, $ \lim_{x\to \infty} \frac 1x \char93 \{n\le x: \gamma_2(n)<x^u\}=
\lim_{x\to \infty} \frac 1x \char93 \{n\le x: n/P(n) < x^u\}
= 1-\rho(1/(1-u))$, where $ \rho$ is the Dickman function. We also show that $ n/P(n)$ can very often be much larger than $ \gamma_2(n)$, namely by proving that, given any $ c\in [1,\xi)$, where $ \xi$ is the unique solution to $ \xi\log 2 = \log(1+\xi)+\xi \log(1+1/\xi)$, then

$\displaystyle \char93 \{n\le x: \gamma_2(n) \ge n/(c\log n)\} = o\left( \char93 \{n\le x: n/P(n) \ge n/(c\log n)\} \right) \qquad (x\to \infty).$


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Received November 24 2011; revised version received January 28 2012. Published in Journal of Integer Sequences, February 5 2012.


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