The germ of this post arises when I was trying to think what terms of the so-called exact form of the prime-counting function satisfy an inequality of the form $\text{term}(x+y)\leq \text{term}(x)+\text{term}(y)$, or of the form $\text{term}(x+y)\leq \text{term}(x)+\text{term}(y)$.

Is in the literature several problems related to the prime-counting function $\pi(x)$, one of the most famous is the Riemann hypothesis. Riemann provide us a formula, called the exact form, that you can see in the section *Exact form* of the Wikipedia *Prime counting function*. But is in the literature other unsolved problem,
a less famous problem the *Second Hardy–Littlewood conjecture*, see the corresponding Wikipedia.

My belief is the following conjecture should be easy to get, since I believe that the mistery of these unsolved problems does not lie in the term that I evoke in the following conjecture, any case I believe that this question is interesting for this site, and I'm curious to know how you analyze the inequality.

**Conjecture.** *For real numbers* $x\geq 2$ *and* $y\geq 2$ *the following inequality holds*

$$\begin{multline} \frac{1}{\pi}\left(\arctan\left(\frac{\pi}{\log x}\right)+\arctan\left(\frac{\pi}{\log y}\right)-\arctan\left(\frac{\pi}{\log (x+y)}\right)\right) \\ \leq\frac{1}{\log x}+\frac{1}{\log y}-\frac{1}{\log (x+y)}. \end{multline}$$

Question.Can you prove previous conjecture?Many thanks.

I know methods to solve inequalities more easy than this.