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The Riemann Zeta function and the Fourier Transform

From Wikipedia:

Riemann did not discuss his hypothesis in any other publication and there is no evidence of private communications in which he claimed to have a proof of this conjecture. Instead, he presented as certain some other results relating to the quantity and arrangement of zeros in the critical strip which have all been proved, with the exception of only one, by other mathematicians in the following years. In particular, Riemann, in addition to giving an estimate of the number of zeros with real part in the interval [0,1] and imaginary part in [-T,T], stated that the fraction of such zeros lying on the critical line tends to 1 when T tends to infinity. Riemann believed he had a rigorous proof of this last statement, which, as he explains in a private communication to a colleague, he did not publish because it was not yet sufficiently simplified. Even today, even this weak form of the hypothesis awaits proof or disprovement.
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Contour plots and the Riemann zeta function

Here are some suggestive images of notable graphs relating to the Riemann zeta function. More precisely, we plot the contour lines of the module of the complex function ζ(x+iy) i.e. |ζ(x+iy)|. For n=15 contours:

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