On the Truncated Kernel Function

p|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 γ2(n) for n ≤ x is (1 + o(1))x/ log x as x → ∞ and that ∑n≤x 1/γ2(n) = (1+ o(1))ηx/ log x, where η = ζ(2)ζ(3)/ζ(6). We further show that, given any positive real number u < 1, limx→∞ 1 x#{n ≤ x : γ2(n) < xu} = limx→∞ 1 x#{n ≤ x : n/P (n) < xu} = 1− ρ(1/(1− u)), where ρ is the Dickman function. We also show that n/P (n) can very often be