Robin inequality for $7-$free integers

Recall that an integer is $t-$free iff it is not divisible by $p^t$ for some prime $p.$ We give a method to check Robin inequality $σ(n) < e^γn\log\log n,$ for $t-$free integers $n$ and apply it for $t=6,7.$ We introduce $Ψ_t,$ a generalization of Dedekind $Ψ$ function defined for any integer $t\ge 2$ by $Ψ_t(n):=n\prod_{p | n}(1+1/p+...+1/p^{t-1}).$ If $n$ is $t-$free then the sum of divisor function $σ(n)$ is $\le Ψ_t(n).$ We characterize the champions for $x \mapsto Ψ_t(x)/x,$ as primorial numbers. Define the ratio $R_t(n):=\frac{Ψ_t(n)}{n\log\log n}.$ We prove that, for all $t$, there exists an integer $n_1(t),$ such that we have $R_t(N_n)< e^γ$ for $n\ge n_1,$ where $N_n=\prod_{k=1}^np_k.$ Further, by combinatorial arguments, this can be extended to $R_t(N)\le e^γ$ for all $N\ge N_n,$ such that $n\ge n_1(t).$ This yields Robin inequality for $t=6,\,7.$ For $t$ varying slowly with $N$, we also derive $R_t(N)< e^γ.$

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