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I am looking for references on the following results. In what follows $\pi(x)$ denotes the prime counting function.

Result 1. For all real $k>1$ there exists $x^k_0 \in \mathbb{R}$ such that for all $x\ge x^k_0$ the following inequality holds, $$\pi(kx)+\pi(x)>2\pi\left(\left(\dfrac{k+1}{2}\right)x\right)$$ Result 2. For all real $k\ge 2$ and $\varepsilon>0$ there exists a prime between $((1+\varepsilon)n+1)^k$ and $n^k$ for all sufficiently large $n$.

I have searched the internet for references on these results. But the closest I found via searching the first was something called the Second Hardy-Littlewood Conjecture. The third seemed to be very close to Legendre's Conjecture. In fact, when these results were shown to an expert, she said that these results are all deducible by elementary methods. That's why I would be glad if someone could point me to some references from which these results can be deduced.

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    $\begingroup$ The second is immediate from prime number theorem. We have a more precise result: for any $C>0$ and large enough $n$, there is a prime between $n$ and $Cn$ (apply this to $C=(1+\varepsilon)^k$. The former you can probably prove using an asymptotic expansion $\pi(x)=x/\log x+x/(\log x)^2+O(x/(\log x)^3)$. $\endgroup$
    – Wojowu
    Commented Aug 10, 2021 at 13:09

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