Section 25.3 Toward the Riemann Hypothesis
![The zeta function on [-10,10]](../images/sageplot-sage-zeta-plot-2.png)
plot(zeta,-10,10,ymax=10,ymin=-1)
)Subsection 25.3.1 Zeta beyond the series
Wait a minute! What was that plot? Shouldn't
graphics_array([complex_plot(zeta, (-20,20), (-20,20)),complex_plot(lambda z: z, (-3,3),(-3,3))])
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zeta(-1)
You may wish to view some dueling videos on this topic at Numberphile, a rebuttal, or another excellent attempt.
Subsection 25.3.2 Zeta on some lines
Let's get a sense for what theplot3d(lambda x,y: abs(zeta(x+i*y)),(0,1),(-20,20), plot_points=100) + plot3d(0,(0,1),(-20,20), color='green',alpha=.5)
.


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var('t')
def _(sig=slider(.01, .99, .01, 0.5, label=r'\(\sigma\)'), end=slider(2,100,1,40, label=r'end of \(t\)')):
p = plot(lambda t: abs(zeta(sig+t*i)), -end,end, rgbcolor=hue(0.7),ymin=0)
q = complex_plot(zeta,(0,.99),(-end,end), aspect_ratio=1/end) + line([(sig,-end),(sig,end)], linestyle='--')
show(graphics_array([p,q]),figsize=[5,3])
Remark 25.3.5.
It is not really possible to fully visualize a complex function of complex input. So we often pick some line in the complex plane, such as where the real part equals 1 (sort of like

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def _(sig=slider(.01, .99, .01, 0.5, label=r'\(\sigma\)')):
end=30
p = parametric_plot((lambda t: zeta(sig+t*i).real(), lambda t: zeta(sig+t*i).imag()), (0,end), rgbcolor=hue(0.7),plot_points=300)
q = complex_plot(zeta,(0,.99),(0,end), aspect_ratio=1/end) + line([(sig,0),(sig,end)], linestyle='--')
show(graphics_array([p,q]), figsize=[5,3])
Conjecture 25.3.7. Riemann Hypothesis.
All the zeros of