File:RiemannCriticalLine.svg

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Summary

Description

English: Graph of real (red) and imaginary (blue) parts of the critical line Re(z)=1/2 of the Riemann zeta function.

Date 20 November 2008

Source

Own work. Made with Mathematica using the following code:

Show[Plot[{Re[Zeta[1/2+I x]], Im[Zeta[1/2+I x]]}, {x,-30, 30},AxesLabel->{"x"} , PlotStyle->{Red, Blue}, Ticks->{Table[4x-28,{x,0,14}]}, ImageSize->{800,600}],
Graphics[Text[Style[\[DoubleStruckCapitalR][\[Zeta][ I x + "1/2"]],14,Red ,Background ->White],{-22,2.6} ]],
Graphics[Text[Style[\[GothicCapitalI][\[Zeta][ I x + "1/2"]],14,Blue ,Background ->White],{-14,2.6} ]]]

Author Slonzor

Permission
(Reusing this file) Public Domain

SVG development

The SVG code is valid.

This plot was created with Matplotlib by Krishnavedala.

Source code

Python code

Source code

import mpmath
import numpy as np
from matplotlib import pyplot as plt
plt.rcParams['svg.fonttype'] = 'path'

x = np.linspace(-30, 30, 300)
y = [complex(1,1)]*len(x)
for p, xx in enumerate(x):
    t = mpmath.nstr(mpmath.mpc(0.5 + xx*1j))
    y[p] = mpmath.zeta(t)

fig = plt.figure(figsize=[13,6])
ax = fig.add_subplot(111)

ax.spines['left'].set_position('zero')
ax.spines['right'].set_color('none')
ax.spines['bottom'].set_position('zero')
ax.spines['top'].set_color('none')
ax.spines['left'].set_smart_bounds(True)
ax.spines['bottom'].set_smart_bounds(True)
ax.xaxis.set_ticks_position('bottom')
ax.yaxis.set_ticks_position('left')

ax.text(-25,2.7, '$\\Re\\left[\\zeta\\left(\\frac{1}{2}+ix\\right)\\right]$', size='xx-large', color='red')
ax.text(-15,2.7, '$\\Im\\left[\\zeta\\left(\\frac{1}{2}+ix\\right)\\right]$', size='xx-large', color='blue')

ax.plot(x, [yy.real for yy in y], label='Real', color='red')
ax.plot(x, [yy.imag for yy in y], label='Imag', color='blue')
# ax.legend(loc=(.6,.8))
ax.minorticks_on()
ax.grid(b=True, which='major', ls='-', lw=1.5)
ax.grid(b=True, which='minor', ls='--', lw=.5)
fig.savefig('RiemannCriticalLine.svg', bbox_inches='tight')

Licensing

I, the copyright holder of this work, release this work into the public domain. This applies worldwide.
In some countries this may not be legally possible; if so:
I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.

File history

Click on a date/time to view the file as it appeared at that time.

	Date/Time	Dimensions	User	Comment
current	20:01, 23 August 2017	933 × 434 (50 KB)	Krishnavedala	much reduced vector version
	22:28, 24 September 2009	800 × 600 (122 KB)	Geek3	linewidth=1px
	19:33, 20 November 2008	800 × 600 (122 KB)	Slonzor	Man i've messed this up a lot of times.
	19:27, 20 November 2008	800 × 600 (3.36 MB)	Slonzor
	19:23, 20 November 2008	800 × 600 (3.36 MB)	Slonzor
	19:18, 20 November 2008	800 × 600 (3.36 MB)	Slonzor
	19:13, 20 November 2008	800 × 600 (79 KB)	Slonzor	{{Information \|Description={{en\|1=Graph of real (red) and imaginary (blue) parts of the critical line Re(z)=1/2 of the Riemann zeta function.}} \|Source=Own work. Made with Mathematica using the following code: <code><nowiki>Show[Plot[{Re[Zeta[1/2+I x]],