zeta_zeroes.sagews

v = [float(a) for a in zeta_zeros()]
t = stats.TimeSeries(v)

%timeit len(t.clip_remove(max=100))

625 loops, best of 3: 256 µs per loop

def zeta_pi(X):
    return len(t.clip_remove(max=X))

zeta_pi(20)

1

def g(n, **kwds):
    p = plot(zeta_pi, 1,n,
         plot_points=1000,rgbcolor='red',
         fillcolor=(.9,.9,.9),fill=True, **kwds)
    return p

g(30, thickness=2)

f(T) = T/(2*pi) * log(T/(2*pi*e))
g(50) + f.plot(1,50)

g(100) + f.plot(1,100)

g(1000) + f.plot(1,1000)

︠fab641af-b6b9-497c-971b-72094f300069︠

︠31487e02-e645-4ba9-93e7-bdf17b569659︠

︠1c52d33c-042e-4f56-a46a-be5cad78d266︠
def g(n, **kwds):
    t = stats.TimeSeries(v)[:n]
    x = [(0,0)]+ [(b,a+1) for a,b in enumerate(t.sums())]
    g = plot_step_function(x, **kwds)
    show(g)

g(5, thickness=3)

g(10, thickness=2)

g(100)

g(1000,thickness=.5)

g(100000)

t = stats.TimeSeries(v)
x = [(0,0)]+ [(b,a+1) for a,b in enumerate(t.sums())]
g = plot_step_function(x)
show(g,gridlines=True)

︠7bf5da7d-7fb4-495f-a920-f6c622b1294f︠
g = stats.TimeSeries(v).plot_histogram(bins=100, normalize=False); g

g.show(frame=True, axes=False)

f(T) = (T/(2*pi)) * log(T/(2*pi*e))
f1 = f.derivative(T)
h = g + (f1 * g.ymax()/f1(v[-1])).plot(2,70000,color='red')
h.show(frame=True, axes=False)

show(f)

$$T \ {\mapsto}\ \frac{T \log\left(\frac{T e^{\left(-1\right)}}{2 \, \pi}\right)}{2 \, \pi}$$

N(f(74920.827498994))

99998.5300884912

N(f(10100)-f(10000))

117.415711316151

N(f(10000))

10142.0903475268

show(f1)

$$T \ {\mapsto}\ \frac{\log\left(\frac{T e^{\left(-1\right)}}{2 \, \pi}\right)}{2 \, \pi} + \frac{1}{2 \, \pi}$$

N(f1(10000))

1.17336397784458

len(v)

100000

v[-1]

74920.827498994

Gaps

t = stats.TimeSeries(v)

t.diffs().mean()

0.7490744184827047

s = t.diffs()

%time w = stats.TimeSeries([s[:i].mean() for i in [5,50,..,len(t)]])

CPU time: 0.52 s, Wall time: 0.52 s

w.plot()

show(t.diffs()[:2000].plot_histogram(bins=100, normalize=False), xmax=2.3, frame=True)

show(t.diffs()[:10000].plot_histogram(bins=100, normalize=False), xmax=2.3, frame=True)

show(t.diffs().plot_histogram(bins=500, normalize=False), xmax=2.3, frame=True, gridlines=True, axes=False)

len(v)

100000

len(t.diffs())

99999