On the Truncated Kernel Function

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

is the largest prime factor of

. 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

, $\lim_{x\to \infty} \frac 1x \char93 \{n\le x: \gamma_2(n)<x^u\}= \lim_{x\to \infty} \frac 1x \char93 \{n\le x: n/P(n) < x^u\} = 1-\rho(1/(1-u))$ , where $\rho$ is the Dickman function. We also show that

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

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

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