CoCalc -- code.sagews

illustrations for RH book

Project: RH -- Prime Numbers and the Riemann Hypothesis

Views: ¹⁸⁵⁷

%load code.sage

%time
%load code.sage
%time draw()
︠5cee7c63-955c-4c55-bf31-df93d0fea32es︠
['_'.join(x.split('_')[1:]) for x in globals() if x.startswith('fig_')]

['primegapdist', 'inverse_of_log', 'pi_riemann_gauss', 'moebius', 'log', 'li_pi_loginv', 'logXoverX', 'pure_tone', 'questions', 'zero_spacing', 'proportion_primes', 'prime_pi_aspect1', 'Psiprime', 'theta_C_intro', 'psi', 'Phi', 'waves', 'mini_phihat_even', 'phi', 'calculus', 'jump', 'primegap_race', 'phihat_even_all', 'prime_pi', 'primes_line', 'staircase_riemann_spectrum', 'simple_staircase', 'even_function', 'fourier_machine', 'mixed_tone', 'multpar', 'riemann_spectrum_gaps', 'aplusone', 'cesaro', 'random_walks', 'Rk', 'Psi', 'simrates', 'sieves', 'two_delta', 'sawtooth', 'dirac', 'prime_pi_nofill', 'oo_integral', 'theta_C', 'psi_waves', 'phihat_even', 'li_minus_pi', 'factor_tree', 'even_pi', 'gaussian_primes', 'erat']

for x in ['li_minus_pi', 'factor_tree', 'even_pi', 'gaussian_primes', 'erat']:
    %time draw(x)
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%load code.sage
draw('phi')

Drawing phi...   (time = 22.0821690559 seconds)

%load code.sage
draw('moebius')

Drawing moebius...   (time = 7.25760006905 seconds)

reset()
︠4a2685c6-fdc3-4cb2-812b-d78f28917eb3s︠
%load code.sage
draw("li_minus_pi")

Drawing li_minus_pi...   (time = 39.5458168983 seconds)

%time
%load code.sage
draw('theta_C_intro')

Drawing theta_C_intro...   (time = 8.55938410759 seconds)
CPU time: 7.83 s, Wall time: 8.77 s

%load code.sage
g = fig_pi_Li(0,0)

︠83dda20b-e4c9-4f8a-bc87-0e174ed8b0cb︠
%time
%load code.sage
draw('gaussian_primes')

Drawing gaussian_primes...   (time = 18.4877309799 seconds)
CPU time: 16.05 s, Wall time: 18.74 s

gaussian_primes(10)

[i + 1, i + 2, 2*i + 1, 3, 3*i + 2, 2*i + 3, i + 4, 4*i + 1, 5*i + 2, 2*i + 5, i + 6, 6*i + 1, 5*i + 4, 4*i + 5, 7, 7*i + 2, 2*i + 7, 6*i + 5, 5*i + 6, 3*i + 8, 8*i + 3, 8*i + 5, 5*i + 8, 9*i + 4, 4*i + 9, i + 10, 10*i + 1, 10*i + 3, 3*i + 10, 8*i + 7, 7*i + 8, 7*i + 10, 10*i + 7, 10*i + 9, 9*i + 10]

K.<i> = QuadraticField(-1)

i^2

-1


︠3b1dda8f-a711-462e-95b9-5e78e8a146c5︠
B = 10
%time v = K.primes_of_bounded_norm(2*B*B)
w = []
for I in v:
    a = I.gens_reduced()[0]
    if abs(a[0])<=B and abs(a[1]) <= B:
        w.extend([a,-a,i*a,-i*a])
w = [z for z in w if z[0]>0 and z[1]>=0]

CPU time: 0.05 s, Wall time: 0.05 s

g.save?
︠1571ab06-a964-4d30-80a9-866cfb019b51︠
%time show(points(w, pointsize=90, zorder=10), aspect_ratio=1, svg=True, figsize=7, fontsize=16, gridlines=True, ticks=[[-1..B],[-1..B]], xmin=-.5, ymin=-.5)

CPU time: 0.30 s, Wall time: 0.72 s

B = 100
%time v = K.primes_of_bounded_norm(2*B*B)
w = []
for I in v:
    a = I.gens_reduced()[0]
    if abs(a[0])<=B and abs(a[1]) <= B:
        w.extend([a,-a,i*a,-i*a])
w = [z for z in w if z[0]>0 and z[1]>=0]

CPU time: 2.71 s, Wall time: 2.71 s

cartesian_product([[1..3], [1..3]])

Error in lines 1-1
Traceback (most recent call last):
  File "/projects/54949eee-57da-4bd7-bb43-c2602b429f9a/.sagemathcloud/sage_server.py", line 873, in execute
    exec compile(block+'\n', '', 'single') in namespace, locals
  File "", line 1, in <module>
  File "/usr/local/sage/sage-6.5/local/lib/python2.7/site-packages/sage/categories/covariant_functorial_construction.py", line 218, in __call__
    assert(all( hasattr(arg, self._functor_name) for arg in args))
AssertionError

p1 = points(w, pointsize=20, zorder=10)
p2 = points(list(cartesian_product_iterator([[0..B],[0..B]])), pointsize=3, color='grey', zorder=15)
%time show(p1+p2, aspect_ratio=1, svg=True, frame=True, axes=False, figsize=10)

CPU time: 4.57 s, Wall time: 9.05 s

︠e2f7abd7-d6ee-4117-8f71-bb5b038917bc︠
draw("cesaro")

Drawing cesaro...   (time = 161.922121048 seconds)

draw('theta_C_intro')

Drawing theta_C_intro...   (time = 41.996740818 seconds)

%load code.sage
draw('psi')

Drawing psi...   (time = 1.65290188789 seconds)

%load code.sage
draw('waves')

Drawing waves...   (time = 4.01206207275 seconds)

%load code.sage
draw('sawtooth')

Drawing sawtooth...   (time = 0.904992103577 seconds)

%load code.sage
draw('pure_tone')
draw('mixed_tone')

Drawing pure_tone...   (time = 0.395801067352 seconds)
Drawing mixed_tone...   (time = 0.772159099579 seconds)

%load code.sage
draw('prime_pi')
draw('prime_pi_aspect1')

Drawing prime_pi...   (time = 13.0921411514 seconds)
Drawing prime_pi_aspect1...   (time = 4.44406914711 seconds)

%load code.sage
for a in 'aplusone calculus jump dirac two_delta'.split():
    draw(a)

Drawing aplusone...   (time = 0.605562925339 seconds)
Drawing calculus...   (time = 0.686944007874 seconds)
Drawing jump...   (time = 2.55016493797 seconds)
Drawing dirac...   (time = 0.0844738483429 seconds)
Drawing two_delta...   (time = 0.150384902954 seconds)

%load code.sage
draw('simple_staircase')

Drawing simple_staircase...   (time = 0.414288043976 seconds)

%load code.sage
draw('even_function')
draw('even_pi')

Drawing even_function...   (time = 0.670686006546 seconds)
Drawing even_pi...   (time = 0.580007076263 seconds)

%load code.sage
draw('mini_phihat_even'),draw('phihat_even')

Drawing mini_phihat_even...   (time = 49.5483169556 seconds)
Drawing phihat_even... 

%load code.sage
draw('theta_C_intro')

%load code.sage
draw('zero_spacing')

plot?
︠1aee8e1e-a26f-45e2-a0e8-241a153aa917︠

︠c81b67a4-6391-4bd3-a743-8c4ba24cd6ab︠

draw('Psiprime')

Drawing Psiprime...   (time = 0.26216506958 seconds)

draw('riemann_spectrum_gaps')

Drawing riemann_spectrum_gaps...   (time = 1.61334013939 seconds)


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︠4402b7b7-4adc-4d16-bf55-1bf2faab6f21︠
def fig_Phi():
    g = line([(0,0),(0,100)], rgbcolor='black')
    xmax = 20
    ymax = 50
    for n in [1..xmax]:
        if is_prime_power(n):
            if n == 1:
                h = log(2*pi)
            else:
                h = log(factor(n)[0][0])
            h /= sqrt(n)
            h = float(h)*50
            print h
            g += arrow((log(n),0),(log(n),h), width=1)
            g += arrow((-log(n),0),(-log(n),h), width=1)
            g += line([(log(n),0),(log(n),100)], color='black', thickness=.3)
            g += line([(-log(n),0),(-log(n),100)], color='black', thickness=.3)
            if n in [2, 5, 16]:
               g += text("log(%s)"%n, (log(n),-5), rgbcolor='black', fontsize=12)
               g += line([(log(n),-2), (log(n),0)], rgbcolor='black')
               g += text("log(%s)"%n, (-log(n),-5), rgbcolor='black', fontsize=12)
               g += line([(-log(n),-2), (-log(n),0)], rgbcolor='black')
    g += line([(-log(xmax)-1,0), (log(xmax)+1,0)], thickness=2)
    return g

show(fig_Phi(), axes=False)


︠8c58f170-89df-4317-9035-cb453164414a︠
%load code.sage

draw('zero_spacing')

Drawing zero_spacing...   (time = 1.18501615524 seconds)

draw('Phi')

Drawing Phi...  91.8938533205
5064535867
7142050299
328679514
9881257777
7742452005
2532267934
3102048111
1496313929
5694477973
664339757
3577584631
7750314621
 (time = 0.299562931061 seconds)

%time
draw("theta_C_intro")

Drawing theta_C_intro...   (time = 1.07501602173 seconds)
CPU time: 1.08 s, Wall time: 1.06 s

draw("cesaro")

Drawing cesaro...   (time = 3.174036026 seconds)

def f(B):
    v = prime_gap_distribution(10^B)
    z = [v[i] if i<len(v) else 0 for i in [2,4,6,8,100,252]]
    print "$10^{%s}$ & "%B + " & ".join([str(a) for a in z]) + r"\\\hline"

for B in [1..8]:
    f(B)

$10^{1}$ & 2 & 0 & 0 & 0 & 0 & 0\\\hline
$10^{2}$ & 8 & 7 & 7 & 1 & 0 & 0\\\hline
$10^{3}$ & 35 & 40 & 44 & 15 & 0 & 0\\\hline
$10^{4}$ & 205 & 202 & 299 & 101 & 0 & 0\\\hline
$10^{5}$ & 1224 & 1215 & 1940 & 773 & 0 & 0\\\hline
$10^{6}$ & 8169 & 8143 & 13549 & 5569 & 2 & 0\\\hline
$10^{7}$ & 58980 & 58621 & 99987 & 42352 & 36 & 0\\\hline
$10^{8}$ & 440312 & 440257 & 768752 & 334180 & 878 & 0\\\hline

%time
f(500*10^6)

[1840170, 1841265, 3257346, 1434059, 0]
CPU time: 92.78 s, Wall time: 87.69 s

%time
draw("theta_C")

Drawing theta_C...   (time = 5.19401979446 seconds)
CPU time: 5.19 s, Wall time: 5.05 s

%time
draw("random_walks")

Drawing random_walks...   (time = 67.0932559967 seconds)
CPU time: 67.09 s, Wall time: 66.59 s










︠59430b3a-b0f2-46ba-93b8-8f51522a2f7b︠
p = area_under_inverse_log(30)

show(p, aspect_ratio='automatic', svg=True)

p.aspect_ratio()

1.0

a = p[0] + p[1] + sum(p[3:]); b = p[2]

b.set_aspect_ratio('automatic')

b.aspect_ratio()

'automatic'

a+b


polygon2d.options

{'aspect_ratio': 1.0, 'alpha': 1, 'fill': True, 'legend_label': None, 'rgbcolor': (0, 0, 1), 'thickness': None}


circle.options

{'edgecolor': 'blue', 'facecolor': 'blue', 'clip': True, 'legend_label': None, 'thickness': 1, 'zorder': 5, 'aspect_ratio': 1.0, 'alpha': 1, 'linestyle': 'solid', 'fill': False}