Dirichlet eta function and the Riemann hypothesis

Question

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?

Answer 1

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.