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
Snapshots
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.
Permanent Citation
"The Riemann Zeta Function in Four Dimensions"
http://demonstrations.wolfram.com/TheRiemannZetaFunctionInFourDimensions/
Wolfram Demonstrations Project
Published: April 5 2011