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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 $k\ge 1$ and for all $\varepsilon \in\left(0,\ln\sqrt{1+\dfrac{1}{k}}\right]$ there exists $M_{(k,\varepsilon)}>0$ such that for all $y\ge M_{(k,\varepsilon)}$ we have, $$\dfrac{y}{\ln y−(1−ϵ)}<π(y)<\dfrac{y}{\ln y−(1+ϵ)}$$ and hence, $$π(ky)+π(y)>π((k+1)y)$$

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

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

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. Assume that for all $k>1$ and $\varepsilon\in (0,\ln \sqrt{2}]$ there exists $y_0\in \mathbb{R}$ (depending on $\varepsilon$ only) such that for all $y\ge y_0$ we have, $$\dfrac{y}{\ln y-(1-\varepsilon)}<\pi(y)<\dfrac{y}{\ln y-(1+\varepsilon)}$$Then, $$\pi(ky)+\pi(y)\ge \pi((k+1)y)$$ for all $y\ge y_0$ where $\pi$ is the Prime Counting Function.

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.

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 $k\ge 1$ and for all $\varepsilon \in\left(0,\ln\sqrt{1+\dfrac{1}{k}}\right]$ there exists $M_{(k,\varepsilon)}>0$ such that for all $y\ge M_{(k,\varepsilon)}$ we have, $$\dfrac{y}{\ln y−(1−ϵ)}<π(y)<\dfrac{y}{\ln y−(1+ϵ)}$$ and hence, $$π(ky)+π(y)>π((k+1)y)$$

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

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 Hrady-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. Assume that for all $k>1$ and $\varepsilon\in (0,\ln \sqrt{2}]$ there exists $y_0\in \mathbb{R}$ (depending on $\varepsilon$ only) such that for all $y\ge y_0$ we have, $$\dfrac{y}{\ln y-(1-\varepsilon)}<\pi(y)<\dfrac{y}{\ln y-(1+\varepsilon)}$$Then, $$\pi(ky)+\pi(y)\ge \pi((k+1)y)$$ for all $y\ge y_0$ where $\pi$ is the Prime Counting Function.

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.

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. Assume that for all $k>1$ and $\varepsilon\in (0,\ln \sqrt{2}]$ there exists $y_0\in \mathbb{R}$ (depending on $\varepsilon$ only) such that for all $y\ge y_0$ we have, $$\dfrac{y}{\ln y-(1-\varepsilon)}<\pi(y)<\dfrac{y}{\ln y-(1+\varepsilon)}$$Then, $$\pi(ky)+\pi(y)\ge \pi((k+1)y)$$ for all $y\ge y_0$ where $\pi$ is the Prime Counting Function.

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.