1
$\begingroup$

I posed a conjecture as follows that is a special case of Second Hardy–Littlewood conjecture:

For $n, x \ge 2$ be two integers then: $$P_{2n} \ge 2P_n$$ and $$\pi(2x) \le 2\pi(x)$$

Where $P_n$ is $n$-$th$ prime and $\pi(x)$ is Prime-counting function

My question: Is the conjecture above contradictory (inconsistent) to the Prime k-tuple conjecture? I am looking for a proof, comment or reference.

$\endgroup$

1 Answer 1

3
$\begingroup$

We have $\pi(2x)<2\pi(x)$ for any $x\geq 11$. For a proof see here. Similarly, it follows from standard bounds that $P_{2n}>2P_n$ when $n$ is sufficiently large.

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.