The Riemann Zeta Function in Four Dimensions

The Riemann zeta function is the analytic continuation of the function , where , . The blue curve is a plot of . You can vary using the slider; it acts as the fourth dimension. The black line marks the origin of the - complex plane. The red arrows mark where the zeta function (blue line) crosses the black line (); these are some of the zeros of the zeta function. The so-called trivial zeros appear at negative even integers when . The Riemann conjecture states that the nontrivial zeros all lie on the critical line .


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See an English translation of [1].
[1] B. Riemann, "Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse," Monatsberichte der Berliner Akademie in Gesammelte Werke, Teubner, Leipzig 1892. http://www.claymath.org/millennium-problems/riemann-hypothesis.
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