If my calculations were rights, it's easy to deduce combining the abc conjecture with Nicolas equivalence to the Riemann's Hypothesis that:
$\qquad\qquad\forall \epsilon>0$ there exists a positive constant $C=C(\epsilon)$ such that $$e^{\gamma}\log \log (N_k\mathcal{N}_k)<\frac{N_k\left(C(\epsilon)\cdot\operatorname{rad}(abc)^{1+\epsilon}-N_k\right)}{\varphi(N_k\mathcal{N}_k)}.$$ Here I am denoting by $a\equiv a_k=N_k$ the primorial of order $k$, and $b\equiv b_k$ (now we are omiting the subscripts $k$ for the variables $a,b$ and $c$), $b=\mathcal{N}_k=\prod_{j=k+1}^{2k}p_j$ with $p_j$ the jth prime number (I choose this definition for $b's$ by symmetry, but one can do different definitions for $b$ as coprime with $a$, I don't know what's can to be more interesting that mine), and $$c=a+b,$$ and thus with $\varphi(n)$ the Euler totient function, $\operatorname{rad}(n)$ the radical of an integer $n\geq 1$ and $\gamma$ the Euler Mascheroni constant.
I believe that these simple calculations were rigths, and I am interested if it's possible do more calculations, I say easy deduction to study such inequality.
Question. A) What's about an unconditional lower bound (I believe that this is the interesting bound, the lower) of $$\operatorname{rad}\left(N_k\mathcal{N}_k(N_k+\mathcal{N}_k)\right)?$$ If this was in the literature provide us information about the references. Can you deduce some about it?
B) And what's about a more general study of the inequality? I am asking about more calculations involving the Euler totient function, and the logarithms in LHS with the purpose to study the veracity of our inequality for integers $k\geq 1$. Many thanks.
References: One can find Nicolas statement in this Wikipedia, subsection 3.7.3, or in his paper Nicolas, Petites valeurs de la fonction d’Euler, J. Number Theory 17 (1983), no. 3. And the statement for the abc conjecture is for example is in the first paragraph of this MathWorld (I say for positive integers in our triples to avoid the absolute values).