On conjectures about the arithmetic function that counts the number of Sophie Germain primes

Question

I've edited this post two years ago on Mathematics Stack Exchange, with identifier 3590406 and same title On conjectures about the arithmetic function that counts the number of Sophie Germain primes, there were some comments from professors (about the veracity of these consejectures). I ask what work can be done to find out a counterexample or if you can to refute some of these conjectures.

In this post we denote (for a fixed positive integer or real number $x$) as $$\operatorname{Germain}(x)=\#\{\text{ primes }p\leq x\,|\,2p+1\text{ is also prime}\}$$ the arithmetic function that counts the number of primes $p$ less than a given positive real $x$ satisfying that $2p+1$ is also prime. As general reference I add the article from Wikipedia Sophie Germain prime that refers that is unproven the existence of infinitely many Sophie Germain primes, and the articles also from Wikipedia Second Hardy–Littlewood conjecture and for Legendre's constant. I was inspired in these articles and a few experiments using Pari/GP scripts to state the following conjectures.

Conjecture 1. One has $$\operatorname{Germain}(x+y)\leq \operatorname{Germain}(x)+\operatorname{Germain}(y)$$ for all integer $x\geq 2$ and all integer $y\geq 2$.

Conjecture 2. One has that $\forall x>87$ $$\operatorname{Germain}(x)<2C\frac{x}{\log^2 x-20}$$ where $C$ denotes the Hardy–Littlewood's twin prime constant.

Question. Are these known, or is it possible to prove or refute any of previous conjectures? Can you find counterexamples for these or add heuristics to know what about the veracity of this kind of conjectures? Many thanks.

I've tested the first conjecture only for the segments of integers $2\leq x,y\leq 500$. The second conjecture is true for the segment $88\leq x\leq 25000$, but my belief is that is false, I've tested different constants $\mu$ in the expression $2C\frac{x}{\log^2 x-\mu}$ that are approximately $\mu\approx 20$. I can add in a comment the scripts written in Pari/GP that I've used to check it.

References:

[1] G. H. Hardy and J. E. Littlewood, Some Problems of 'Partitio Numerorum.' III. On the Expression of a Number as a Sum of Primes, Acta Math. (44), J. E. (1923) pp. 1–70.

Answer 1

A reasonable conjecture is that

$$ \operatorname{Germain}(x) = 2 C \int_2^x \frac{dy}{\log^2 y} + O( x^{1-\delta} )$$ for some $\delta >0$. This is the Hardy-Littlewood conjecture with power-saving error term.

A function-field analogue of this conjecture follows from the methods of my paper On the Chowla and twin primes conjectures over $\mathbb F_q[t]$ with Mark Shusterman.

A good way to test the validity of these conjectures is to see how they relate to this asymptotic. For example, Conjecture 1 follows as long as $x$ and $y$ are not too far apart.

The integral $\int_{2^x} \frac{dx}{\log^2 x}$ has an asymptotic expansion $$x /(\log x)^2 + 2x / (\log x)^3 + 6 x / (\log x)^4+ \dots= x / (\log^2 x - 2 \log x + O(1))$$ and the power savings error term is, once we put it on the denominator, smaller than that $O(1)$, thus it is reasonable to conjecture that

$$\operatorname{Germain}(x) = 2 C \frac{x}{ \log^2 x - 2 \log x + O(1) } $$

and one could even look for numerical evidence about how large this $O(1)$ should be.

So indeed Conjecture 2 should be wrong but a modified version with the $-2 \log x$ could well be correct.