The Robin Inequality for 7-Free Integers
Résumé
Recall that an integer is t-free if and only if it is not divisible by pt for some
prime p. We give a method to check Robin’s inequality, !(n) < e!n log log n, for
t-free integers n and apply it for t = 6, 7. We introduce !t, a generalization of the
Dedekind ! function defined for any integer t ! 2, by
!t(n) := n
!
p|n
"
1 + 1/p + · · · + 1/pt−1#
.
If n is t-free then the sum of divisor function !(n) is " !t(n). We characterize
the champions for x #$ !t(x)/x, as primorial numbers. Define the ratio Rt(n) :=
!t(n)
n log log n. We prove that, for all t, there exists an integer n1(t), such that we have
Rt(Nn) < e! for n ! n1, where Nn =
$n
k=1 pk. Further, by combinatorial arguments,
this can be extended to Rt(N) " e! for all N ! Nn, such that n ! n1(t).
This yields Robin’s inequality for t = 6, 7. For t varying slowly with N, we also
derive Rt(N) < e!.