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.

363012and titleOn a conjecture about the arithmetic function that counts the number of twin primesI add this comment if you can to read or contibute to it. $\endgroup$Conjecture 2seemed false to me. If you say the opposite sure that you're right professor @WillSawin Feel free to expand your comment as an answer, sure that it is excellent. $\endgroup$