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I am interested if possible in $\beta = \frac{2}{3}, \alpha=\frac{3}{2}$ and $z$ is a positive integer or real number. My choice here is related to some progress I make in additive combinatorics (see my last answer to this MO question). It would make my life easier if this was true.

However, I can be less picky: all I really need it seems, is $\alpha < 2$. The closest $\alpha$ is to $2$, of course the more likely the answer to my question could be positive, but it also makes some arguments in my previous MO question less likely to work out. For $\beta$, you can pick up any positive value that would give a positive answer to my question.

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    $\begingroup$ Of course, the answer is "no" for $\alpha\leq 1$. For $\alpha>1$ this is conjecturally true, but not proven. This is a question about maximal prime gaps, specifically whether they are $O(p_n^{\alpha-1})$. See here for known results. $\endgroup$
    – Wojowu
    Commented Jun 21, 2020 at 23:33
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    $\begingroup$ Indeed, it's not even known whether there's always a prime between consecutive squares, so the question here seems hopeless. $\endgroup$
    – Gerry Myerson
    Commented Jun 22, 2020 at 7:27
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    $\begingroup$ Correction from my previous comment: the exponent should be, I think, $\frac{\alpha-1}{\alpha}$, not $\alpha-1$. $\endgroup$
    – Wojowu
    Commented Jun 22, 2020 at 9:33
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    $\begingroup$ Thank you. I guess I will stop wasting my time trying to prove my problem. That was supposed to be the easiest of the two big challenges I am facing, and it turns out to be unproven yet. $\endgroup$
    – Vincent Granville
    Commented Jun 22, 2020 at 13:48

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