I wondered if it is possible to prove or refute the following conjectures involving finite products of Ramanujan primes that are denoted in this post as $R_j$, I add as reference the Wikipedia Ramanujan prime. To create these inequalities I was inspired in Nicolas criterion for the Riemann hypothesis (see [1]).
Definition. For each integer $k\geq 1$ we denote as $$\mathcal{R}_k=\prod_{j=1}^k R_j$$ the product of the first $k$ Ramanujan primes $R_j$.
For example $\mathcal{R}_3=2\cdot 11\cdot 17=374$.
Also I denote the Euler's totient function as $\varphi(n)$.
From previous definitions I was inspired in Nicolas criterion, and inspired in the computational evidence of a Pari/GP script I've stated the following conjectures.
Conjecture 1. For all integer $k>116$ we've that the inequality $$\frac{\mathcal{R}_k}{\varphi(\mathcal{R}_k)}<\frac{1}{2}\log\log \mathcal{R}_k\tag{1}$$ holds.
Conjecture 2. For all integer $k\geq 1$ we've that the inequality $$\frac{1}{3}\log\log \mathcal{R}_k <\frac{\mathcal{R}_k}{\varphi(\mathcal{R}_k)}\tag{2}$$ holds.
Question (Edited). How to prove the first conjecture? What is a reasoning or counterexample disproving the second conjecture? If these conjectures are difficult to elucidate I'm asking about what work can be done about it. Many thanks.
We need to estimate $\mathcal{R}_k$ and $\prod_{1\leq j\leq k}\left(1-\frac{1}{R_j}\right)^{-1}$ (it is well-known that Euler's totient function is multiplicative). The first conjecture doesn't seem a sharp inequality, I believe that maybe there are counterexamples for the second conjecture since seems to me very sharp for the first few thousands of integers $k$'s.
References:
[1] Jean-Louis Nicolas, Petites valeurs de la fonction d’Euler, J. Number Theory 17 (1983), no. 3, pp. 375–388.