def gamma_R(s):
return pi^(-s/2) * gamma(s/2)
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gamma_R(3+I)
pi^(-1/2*I - 3/2)*gamma(1/2*I + 3/2) pi^(-1/2*I - 3/2)*gamma(1/2*I + 3/2) |
z = complex(1/2+10*I)
timeit('psi(z)')
625 loops, best of 3: 496 µs per loop 625 loops, best of 3: 496 µs per loop |
gamma(CDF(1/2-10*I))
3.37872437623e-07 - 1.68936983904e-07*I 3.37872437623e-07 - 1.68936983904e-07*I |
psi1
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\sum_p \frac{\log(p)}{\sqrt{p}} \cdot\cos(\log(p)t)
var('t')
f(t) = -sum(log(p)/sqrt(p)*cos(log(p)*t) for p in primes(10000))
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A = plot(f, 2,100,plot_points=2000) + sum([line([(x,-1),(x,15)],color='red') for x in zeta_zeros()[:100] if x < 100])
A.show(figsize=[18,6],ymin=0)
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Let \pi(x) be the prime counting function.
Take \psi(x)=e^{-t/2}\pi'(e^t), with even extension.
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f(t) = -sum((1/sqrt(p))*cos(log(p)*t) for p in primes(1000))
A = plot(f, 2,100,plot_points=2000) + sum([line([(x,-1),(x,15)],color='red') for x in zeta_zeros()[:100] if x < 100])
A.show(figsize=[18,3],ymin=-3)
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%time
f(t) = -sum(cos(log(p)*t)/sqrt(p) for p in primes(10000))
f = f._fast_float_(t)
CPU time: 10.25 s, Wall time: 10.25 s CPU time: 10.25 s, Wall time: 10.25 s |
B=200
A = plot(f, 2, B, plot_points=2000) + sum([line([(x,-1),(x,15)],color='red') for x in zeta_zeros()[:B] if x < B])
A.show(figsize=[18,3],ymin=-3)
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A.show(figsize=[100,3],ymin=-3)
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%time
g(t) = -sum(cos(theta*log(t)) for theta in zeta_zeros()[:2000])
g = g._fast_float_(t)
CPU time: 1.86 s, Wall time: 1.86 s CPU time: 1.86 s, Wall time: 1.86 s |
B = 1000
A = plot(g, 2, B, plot_points=2000) + sum([line([(x,-1),(x,15)],color='red') for x in primes(B) if x < B])
A.show(figsize=[100,3],ymin=-3)
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%time
f(t) = -sum(cos(log(p)*t)/sqrt(p) for p in primes(10000))
f = f._fast_float_(t)
CPU time: 10.41 s, Wall time: 10.42 s CPU time: 10.41 s, Wall time: 10.42 s |
B = 200
A = plot(f, 2, B, plot_points=2000) + sum([line([(x,-1),(x,15)],color='red') for x in zeta_zeros()[:B] if x < B])
A.show(figsize=[100,3],ymin=-3)
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