The Riemann Zeta Function in Four Dimensions

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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 .

Contributed by: Biswaroop Mukherjee (April 2011)
Open content licensed under CC BY-NC-SA


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Details

See an English translation of [1].

Reference

[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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