def test_zetazero():
cases = [\
(399999999, 156762524.6750591511),
(241389216, 97490234.2276711795),
(526196239, 202950727.691229534),
(542964976, 209039046.578535272),
(1048449112, 388858885.231056486),
(1048449113, 388858885.384337406),
(1048449114, 388858886.002285122),
(1048449115, 388858886.00239369),
(1048449116, 388858886.690745053)
]
for n, v in cases:
print(n, v)
t1 = clock()
ok = zetazero(n).ae(complex(0.5, v))
t2 = clock()
print("ok =", ok, ("(time = %s)" % round(t2 - t1, 3)))
print("Now computing two huge zeros (this may take hours)")
print("Computing zetazero(8637740722917)")
ok = zetazero(8637740722917).ae(complex(0.5, 2124447368584.39296466152))
print("ok =", ok)
ok = zetazero(8637740722918).ae(complex(0.5, 2124447368584.39298170604))
print("ok =", ok)
def test_zetazero():
cases = [\
(399999999, 156762524.6750591511),
(241389216, 97490234.2276711795),
(526196239, 202950727.691229534),
(542964976, 209039046.578535272),
(1048449112, 388858885.231056486),
(1048449113, 388858885.384337406),
(1048449114, 388858886.002285122),
(1048449115, 388858886.00239369),
(1048449116, 388858886.690745053)
]
for n, v in cases:
print n, v,
t1 = clock()
ok = zetazero(n).ae(complex(0.5,v))
t2 = clock()
print "ok =", ok, ("(time = %s)" % round(t2-t1,3))
print "Now computing two huge zeros (this may take hours)"
print "Computing zetazero(8637740722917)"
ok = zetazero(8637740722917).ae(complex(0.5,2124447368584.39296466152))
print "ok =", ok
ok = zetazero(8637740722918).ae(complex(0.5,2124447368584.39298170604))
print "ok =", ok
def calc_zeta(re, img_name):
X, Y, Z = [], [], []
fig = plt.figure()
ax = fig.add_subplot(111)
for i in np.arange(0.1, 50.0, 0.1):
compl = zeta(complex(re, i))
X.append(compl.real)
Y.append(compl.imag)
Z.append(i)
ax.grid(True)
ax.plot(Z, X, label='Im(v)', lw=0.8)
ax.plot(Z, Y, label='Re(v)', lw=0.8)
ax.set_title("Riemann Zeta function - re(s)=%.3f" % re)
ax.set_xlabel("Im(s)")
ax.set_ylabel("Re(v) and Im(v)")
leg = ax.legend(shadow=True)
for t in leg.get_texts():
t.set_fontsize('small')
for l in leg.get_lines():
l.set_linewidth(2.0)
# Plot the zeroes of zeta
for i in range(1, 11):
zero = zetazero(i)
ax.plot(zero.imag, [0.0], "ro")
# Comment this line for autoscale
ax.set_ylim(12, -12)
plt.savefig(img_name)
print("Plot %s !" % img_name)
plt.close()
def getNthZetaZero( n ):
return zetazero( int( n ) )
imag = []
for k in t:
real.append(zeta(complex(0.5, k)).real)
imag.append(zeta(complex(0.5, k)).imag)
fig_size = plt.rcParams["figure.figsize"]
fig_size[0] = 13
fig_size[1] = 6
plt.rcParams["figure.figsize"] = fig_size
plt.axhline(0, color='black', linewidth=1)
plt.axvline(0, color='black', linewidth=1)
for k in [1,2]:
zero = zetazero(k).imag
plt.axvline(zero, color='grey', alpha=1, linewidth=1, linestyle='dashed')
plt.axvline(zero*(-1), color='grey', alpha=1, linewidth=1, linestyle='dashed')
plt.plot(t, real, color='blue', linewidth=3, label=r'$\operatorname{Re}\left(\zeta\left(\frac{1}{2} + i t\right)\right)$')
plt.plot(t, imag, color='red', linewidth=3, label=r'$\operatorname{Im}\left(\zeta\left(\frac{1}{2} + i t\right)\right)$')
plt.xlim(-24,24)
plt.ylim(-3,3)
plt.xlabel("t")
plt.ylabel(r"$\zeta\left(\frac{1}{2} + i t\right)$")
plt.legend(loc='best')
plt.grid(True)
pylab.savefig("critical-line.pdf")
# zeta-zeroes-info.py zeichnet eine komplexe Zahlenebene, in der
# die Nullstellen der Zetafunktion dargestellt werden.
fix, ax = plt.subplots()
re_trivial = []
im_trivial = []
re_nontrivial = []
im_nontrivial = []
for k in range(0, 2):
re_trivial.append(-2 * (k + 1))
im_trivial.append(0)
for j in range(0, 6):
re_nontrivial.append(zetazero(j + 1).real)
im_nontrivial.append(zetazero(j + 1).imag)
re_nontrivial.append(zetazero(j + 1).real)
im_nontrivial.append(-zetazero(j + 1).imag)
plt.axhline(0, color='black', alpha=1, linewidth=1)
plt.axvline(0, color='black', alpha=1, linewidth=1)
plt.axvline(0.5,
color='black',
alpha=1,
linewidth=1,
linestyle='dashed',
label=r'Kritische Gerade $\operatorname{Re}(s)=\frac{1}{2}$')
ax.add_patch(
patches.Rectangle(
(0, -40),
The main objective of plotting RZ function is to verify is all the non-trivial
zeros of RZ function has the real part of 0.5. The locations of non-trivial
zeros are the locations of the prime numbers. Hence, this plot plots the
distribution of prime numbers on the imaginary numberspace.
"""
import matplotlib.pyplot as plt
import mpmath
n = int(raw_input("Input total number of Reimann Zeros to chart [default = 25]") or 25)
listx, r, im, counter = [], [], [], []
for i in range (1,n):
x = mpmath.zetazero(i)
listx.append(x)
ri = x.real
imi = x.imag
r.append(ri)
im.append(imi)
counter.append(i)
zet = plt.figure()
zetx = zet.add_subplot(111)
zetx.scatter(r, im, lw = 2, marker = "+")
zetx.set_xlim([-1, 1])
zetx.grid(True)
plt.title("Distribution of Prime Numbers on Riemann Zeta's critical strip.")
plt.show()
