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On a graph, the nontrivial zeros of the zeta function are on the critical strip. Because the critical strip is vertical, how can any value on the strip be a zero of the zeta function if it isn't directly on the x-axis? For example, how can one of the zeros, (1/2)+(14.13...)i be a zero if it's above the x-axis? Thanks!

Dietrich Burde
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OpieDopee
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    Why not? Where is the problem? – Hagen von Eitzen Sep 02 '13 at 20:36
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    Before one learns about complex numbers, real functions are graphed via the relation $y=f(x)$. With complex numbers, points in the plane are all inputs of the function. The domain is the plane, and the codomain is also the plane. If you wanted to graph such a thing, you would need four dimensions (in practice, dimensions can be color, hue, time etc.) | Others: While amusing, I don't think that not knowing about complex numbers deserves a downvote. – anon Sep 02 '13 at 20:43
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    http://math.stackexchange.com/questions/447027/orientation-of-zeta-zeroes-on-the-critical-line/447253#447253 –  Sep 02 '13 at 20:57
  • Thanks @Andrew ! :-) – Raymond Manzoni Sep 02 '13 at 20:58
  • @RaymondManzoni My pleasure - a great answer. –  Sep 02 '13 at 21:04

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The complex plane has two directions, not only the $x$-axis. If $s=x+iy$ is a zero of the Zeta function, this means $0=\zeta(s)=\zeta(x+iy)$, but $y$ need not be zero (and in fact, is not zero for the nontrivial zeros).

Dietrich Burde
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  • So the complex plane is entirely separate from the real plane? Could it essentially be said that the values for the complex plane would be the x in a f(x) function, but this time it applies for s in the zeta function? Pardon my ignorance on the subject, I need to do more researching. – OpieDopee Sep 02 '13 at 21:43
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    @Ethan: the complex plane is generated by the real axis ($x$) and the imaginary axis ($iy$) and thus made of complex values $s=x+iy$. What is your real plane ? Is this the graph of a $\mathbb{R}\to \mathbb{R}$ function? Here we are concerned with a function $\zeta$ from $\mathbb{C}\to \mathbb{C}$ i.e. from a complex plane to a complex plane. The zeros of $\zeta(s)$ are the points (complex values $s$) from the initial plane that have the image $\zeta(s)=0$ in the destination complex plane. – Raymond Manzoni Sep 02 '13 at 22:13