I recently stumbled upon the following article:

This article claims to prove that the values of the Riemann Zeta function at all odd positive integers are irrational. Last time I checked, this problem was categorized as "near to impossible with the current state of number theory". Naturally, I am a bit skeptic on whether this proof is correct. However, I do not have the time (or more precisely the required skill) to verify the proof by myself.

Is the proof in the above paper correct?

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    $\begingroup$ For something so consequential it is a surprisingly short proof, but this would still take at least a full afternoon for a number theorist to check in detail. At least a few will need to do so before we know the proof is sound. Having said that, Apery's proof on the irrationality of $\zeta(3)$ was important enough, so this would be earth-shattering if it turns out to be correct. I myself am tangentially familiar with this area and did my dissertation in analytic number theory, and in a quick scan nothing leaps out as being grossly incorrect. $\endgroup$ – Sputnik May 28 '18 at 13:16
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    $\begingroup$ Is lemma 3.1 correct? Shouldn't it be an implication only in one direction? $\endgroup$ – Jakobian May 28 '18 at 13:30
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    $\begingroup$ See arxiv.org/abs/1608.03174v4 . The author comments "An erroneous claim of irrationality of all zeta(2n+1) has been withdrawn, with apologies". $\endgroup$ – Lord Shark the Unknown May 28 '18 at 13:31
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    $\begingroup$ @Adam I wondered about that and thought I was reading it wrong... You're right, not only does the implication only work forwards, I believe there is a 0 missing, as in $0<|p\alpha - q| < \varepsilon$. Otherwise the statement is false, as it works for rationals too. Re: arXiv, glad that's cleared up. I thought it was strange that it was dated from 2016 and nothing else had been said about it... $\endgroup$ – Sputnik May 28 '18 at 13:43
  • $\begingroup$ @LordSharktheUnknown Thank you for the link, that clears it up. $\endgroup$ – Klangen May 28 '18 at 13:48

As shown in the comment section, the claim was withdrawn by the author.


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