236
20 Riemann Zeta Function
0
5
10
15
20
25
30
35
40
45
50
t
0
1
2
3
4
| |
-2
-1
0
1
2
3
Fig. 20.1 Absolute value of the Riemann ζ(s)-function for s =
1
2 + i · t, t = 0 · · · 50. The line is
color coded with the angle of s
Figure 20.1 shows the absolute vales of the ζ -function along the strip s =
1
2 + i · t color coded with the angle of ζ . The figure was created with the program
visuzero:
% computing the zeros between 14 ... 50
obj = riemzetaroot(14,50);
t = linspace(0,50,200);
% t values for visualization
t = [t,obj.t.’];
t = sort(t);
s = 1/2 + i * t;
% zeta function argument
obj = riemzeta(s);
% evaluation of the zeta function
% visualization
surface([t;t],[abs(obj.zeta);abs(obj.zeta)],...
[zeros(size(t));zeros(size(t))],...
[angle(obj.zeta);angle(obj.zeta)],...
’facecolor’,’none’, ’edgecolor’, ’flat’,...
’edgelighting’,’phong’, ’linewidth’, 2), shg
xlabel(’t’), ylabel(’|\zeta|’), colorbar
References
1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover Pub., New York
(1972)
2. Gourdon X., Sebah, P.: Numerical evaluation of the Riemann zeta-function (2019). http://
numbers.computation.free.fr/Constants/Miscellaneous/zetaevaluations.pdf
3. Gradstein, I.S., Rhysik, I.M.: Tafeln · Tables II. Verlag Harry Deutsch Thun, Frankfurt (1981)
4. Gutzwiller, M.C.: Chaos in Classical and Quantum Mechanics. Springer, Berlin (1990)
5. Wikipedia. Liste nicht-trivialer Nullstellen der Riemannschen Zetafunktion. https://de.
wikipedia.org/wiki/Liste_nicht-trivialer_Nullstellen_der_Riemannschen_Zetafunktion
6. Zwiebach, B.: A First Course in String Theory. Cambridge University Press, Cambridge (2009)