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I understand that the following relation exists:

$\eta(s) = \left(1-2^{1-s}\right) \zeta(s)$

and from this question:

What are the hypothetical zeros of the Dirichlet eta function

that the Dirichlet eta function zeros "are precisely the zeros of the zeta function", but then I am confused by this paper:

https://arxiv.org/abs/1201.1810

The author claims to "show that the Dirichlet eta function has no zeros in the critical strip off the critical line, consistent with the Riemann hypothesis."

My question is how is this only consistent with the Riemann hypothesis, or in other words, how does this result not imply the Riemann hypothesis?

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  • $\begingroup$ It is consistent in that it proves any zeros in the critical strip must be on the critical line...which is not contradictory to the Riemann hypothesis. $\endgroup$
    – Simply Beautiful Art
    Commented Apr 20, 2017 at 0:29
  • $\begingroup$ @SimplyBeautifulArt Yes, I see how it is consistent. I am confused how it is not also sufficient to prove the Riemann hypothesis. $\endgroup$
    – nellapizza
    Commented Apr 20, 2017 at 1:04
  • $\begingroup$ Note that it does not prove that there cannot be zeros outside of the critical strip, with the exception of the negative even integers. $\endgroup$
    – Simply Beautiful Art
    Commented Apr 20, 2017 at 1:08
  • $\begingroup$ @SimplyBeautifulArt But then from Wolfram MathWorld : "All nontrivial zeros (i.e., those not at negative even integers) of the Riemann zeta function lie inside this strip." Since Dirichlet eta function zeros and Riemann zeta function zeros are the same inside the critical strip and all Dirichlet eta function zeros inside the strip are on the critical line doesn't that imply that all Riemann zeta function zeros are on the critical line as well. $\endgroup$
    – nellapizza
    Commented Apr 20, 2017 at 1:22
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    $\begingroup$ @nellapizza, you are correct that this result implies the Riemann hypothesis. The paper itself is junk. $\endgroup$
    – Peter Humphries
    Commented Apr 20, 2017 at 5:29

1 Answer 1

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The zeroes of $\zeta(s)$ within the critical strip are precisely the same zeroes as of $\eta(s)$ within the critical strip. However, the paper you reference is junk.

I should note that there is a big clue that this was a junk paper: it appeared in the General Mathematics section of the arXiv, and not the Number Theory section. The General Mathematics (and History and Overview) sections of the arXiv have lower standards than the rest, and so have a disproportionately large number of crank papers.

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  • $\begingroup$ In particular about the Riemann hypothesis $\endgroup$
    – reuns
    Commented Apr 21, 2017 at 1:02

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