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\vskip 1cm{\LARGE\bf On the Truncated Kernel Function
}
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\large
Jean-Marie De Koninck \\
D\'epartement de Math\'ematiques et de Statistique  \\
Universit\'e Laval  \\
Qu\'ebec G1V 0A6 \\
Canada \\
\href{mailto:jmdk@mat.ulaval.ca}{\tt jmdk@mat.ulaval.ca}  \\
\ \\
Isma\"{\i}la Diouf \\
D\'epartement de Math\'ematiques et d'Informatique \\
FST - Universit\'e Cheikh Anta DIOP  \\
BP 5005, Dakar-Fann  \\
Senegal  \\
\href{mailto:isma.diouf@gmail.com}{\tt isma.diouf@gmail.com}   \\
\ \\
Nicolas Doyon \\
D\'epartement de Math\'ematiques et de Statistique  \\
Universit\'e Laval 
Qu\'ebec G1V 0A6 \\
Canada \\
\href{mailto:nicodoyon77@hotmail.com}{\tt nicodoyon77@hotmail.com}\\
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\begin{abstract}
We study properties of the {\it truncated kernel function} $\gamma_2$ defined on integers $n\ge 2$ by $\gamma_2(n)=\gamma(n)/P(n)$, where $\gamma(n)=\prod_{p|n}p$ is the well-known {\it 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 $u<1$,
$\lim_{x\to \infty} \frac 1x \#\{n\le x: \gamma_2(n)<x^u\}=
\lim_{x\to \infty} \frac 1x \#\{n\le x: n/P(n) < x^u\}
 = 1-\rho(1/(1-u))$, 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
$$\#\{n\le x: \gamma_2(n) \ge n/(c\log n)\} = o\left( \#\{n\le x: n/P(n) \ge n/(c\log n)\}  \right) \qquad (x\to \infty).$$
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