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Let $p$ be a prime. Then it has been proven by Erdős that $$\sum_{p\mid 2^{n}-1}\frac1{p} \ll \log \log \log n.$$ Let $c<1$ be a positive constant. Then can we prove that $$\sum_{p\mid 2^{n}-1}\frac1{p^{c}}\ll \frac{\log ^{1-c} n}{\log \log n}?$$

Thanks in advance for any assistance.

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    $\begingroup$ Why does it seem to you that the argument should apply? Is it a homework perchance? $\endgroup$
    – Boris Bukh
    Commented Feb 28 at 12:53
  • $\begingroup$ I apologise for the comment. I had some arguments in mind that I thought would translate. But those have a flaw. This is a problem I need to tackle in my research. $\endgroup$
    – Hhhhhhhhhhh
    Commented Mar 6 at 4:50
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    $\begingroup$ Probably this can be obtained using Abel's summation formula. $\endgroup$
    – Konstantinos Gaitanas
    Commented Mar 6 at 8:51

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