Is the following weak version of second Hardy-Littlewood conjecture already known?

Question

Very recently I was going through my previous MSE posts and I stumbled upon some of them regarding the Second Hardy-Littlewood Conjecture which states that,

For all $x,y\ge 2$ we have, $$\pi(x)+\pi(y)\ge \pi(x+y)$$where $\pi$ is the Prime Counting Function.

Observing that the Second Hardy-Littlewood Conjecture is equivalent to the following,

For all $k\ge 1$ and $y\in \mathbb{R}$ the following holds, $$\pi(ky)+\pi(y)\ge \pi((k+1)y)$$where $\pi$ is the Prime Counting Function.

I was wondering whether proving the following weak version of the conjecture is something significant or is it already known,

Proposition. For all $k\ge 1$ there exists $M_{k}>0$ such that for all $y\ge M_{k}$ we have, $$π(ky)+π(y)>π((k+1)y)$$

I searched the internet for something similar to this but even though I found some results closely related to the above, I could't find this exact result.

So my questions are,

• Is the above proposition well-known? If so can anyone point me out to the paper/book that contains a proof of it or even better to some theorem from which this result follows?

• If not then is the proof of this result considered a significant result?

I agree that the second question may seem to be more opinionated but currently this is the best version that I can come up with. I will be glad to receive constructive suggestions on making this question more specific and answerable

Answer 1

I have not seen your proposition before, but it follows routinely from the prime number theorem.

Indeed, there exists an absolute constant $c>0$ such that $$\pi(ky)+\pi(y)-\pi((k+1)y)=\mathrm{Li}(ky)+\mathrm{Li}(y)-\mathrm{Li}((k+1)y)+O_k\left(y e^{-c\sqrt{\log y}}\right),$$ where $$\mathrm{Li}(ky)+\mathrm{Li}(y)-\mathrm{Li}((k+1)y)=\int_0^y\left(\frac{k}{\log(kt)}+\frac{1}{\log t}-\frac{k+1}{\log(k+1)t}\right)dt.$$ For fixed $k$ and $t\to\infty$, the integrand is \begin{align*}&\frac{k}{\log t}\left(1-\frac{\log k+o(1)}{\log t}\right)+\frac{1}{\log t}-\frac{k+1}{\log t}\left(1-\frac{\log(k+1)+o(1)}{\log t}\right)\\[8pt]&=\frac{(k+1)\log(k+1)-k\log k+o(1)}{\log^2 t},\end{align*} whence there exists $C_k$ such that $$\mathrm{Li}(ky)+\mathrm{Li}(y)-\mathrm{Li}((k+1)y)\gg_k\frac{y}{\log^2 y},\qquad y\geq C_k.$$ As $\log^2 y$ grows much slower than $e^{c\sqrt{\log y}}$, we conclude that there exists $M_k$ such that $$\pi(ky)+\pi(y)-\pi((k+1)y)\gg_k\frac{y}{\log^2 y},\qquad y\geq M_k.$$