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If $\frac{\zeta(2 s)}{\zeta(s)}$ is a real number, then must $s$ be real ?

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Of course not. For example, let $2s$ be any non-real zero of the zeta function where $s$ is not a zero.
Through such a point, there will be a curve along which $\zeta(2z)/\zeta(z)$ is real.

EDIT: Here is a plot of the curve where $\zeta(2s)/\zeta(s)$ is real in one region of the plane. A dot indicates the one non-real point in this region where $\zeta(2s) = 0$.

enter image description here

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  • $\begingroup$ And when that ratio is real but non-null? $\endgroup$ – GoldSoundz May 1 at 12:52
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    $\begingroup$ What about it? Everywhere on the curve except at $s$ where $\zeta(2s)=0$, the ratio is real and nonzero. $\endgroup$ – Robert Israel May 1 at 14:53

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