In this post we denote the arithmetic function that counts the number of Ramanujan $R$ primes less than $x$ as $$\pi_{_{R}}(x)=\#\{\text{Ramanujan primes }R\text{ such that }R\leq x\}$$ thus is the analogous for Ramanujan primes than the prime-counting function. Wikipedia has an article for Ramanujan prime. A well-known conjecture in analytic number theory is the known as Second Hardy–Littlewood conjecture (the link also from Wikipedia, where were added references).
I hope that my question* is on topic, I am asking what about the sign of the following function or numerical support, general feedback about the sign of $$f(x,y)=\pi_{_{R}}(x)+\pi_{_{R}}(y)-\pi_{_{R}}(x+y)$$ for integers numbers $x\leq y$ (both integers) such that $x\geq x_0$ for a suitable constant $x_0\geq 2$.
Question. I would like to know some feedback about the sign of $f(x,y)$ when the variables $x$ and $y$ run over positive integers $x_0\leq x$ and $x\leq y\leq N$ for a suitable and integer $x_0$ (if there is some interesting $x_0$, I don't know how oscillates our function) and $N$ also integer, a enough large integer $N$. I mean if you know how to provide me a theoretical reasoning, heuristic or well if you know it from the literature that answer this feedback as a reference request (if it is in the literature I try to search and read it from your recommentation). Many thanks.
Optionally (or alternatively if previous question isn't available because it is very difficult to get a reasoning about it), can you provide numerical evidence for the sign (or oscillations of the sign) of $f(x,y)$ for some $N$ for our lattice of intergers $x_0\leq x\leq y$ and $x\leq y\leq N$, to emphasize $x,y\in\mathbb{Z}_{>0}$ and $x_0$ is a suitable positive integer?
I know just the more humble and simple examples.
Examples. One has that $$3=\pi_{_{R}}(17)=\pi_{_{R}}(8+9)<\pi_{_{R}}(8)+\pi_{_{R}}(9)=2+2=4.$$ Other example is $$7=\pi_{_{R}}(64)=\pi_{_{R}}(17+47)<\pi_{_{R}}(17)+\pi_{_{R}}(47)=3+6=9.$$
*My problem is that I've tried to check what about a Second Hardy–Littlewood conjecture for the arithmetic function $\pi_{_{R}}(x)$ using a programming language but I had difficulties (I know and I can to test the Second Hardy–Littlewood conjecture for the classical prime-counting function and for a given finite segment(s) of integers but in our case, the calculations with a computer that I need to know to get certain computation evidence involving $\pi_{_{R}}(x)$, I had more difficulties).