Section 25.1 Taking the PNT Further
Recall Gauss' approximating function for
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def _(n=(1000,(1000,10^6))):
P = plot(prime_pi,n-1000,n, color='black', legend_label=r'$\pi(x)$')
P += plot(Li,n-1000,n, color='green', legend_label='$Li(x)$')
P += plot(lambda x: Li(x) - .5*Li(sqrt(x)), n-1000,n, color='red', legend_label=r'$Li(x)-\frac{1}{2}Li(\sqrt{x})$')
show(P)
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def _(n=(1000,(1000,10^6)),k=(3,[1..10])):
P = plot(prime_pi,n-1000,n, color='black', legend_label=r'$\pi(x)$')
P += plot(Li,n-1000,n, color='green', legend_label='$Li(x)$')
F = lambda x: sum([Li(x^(1/j))*moebius(j)/j for j in [1..k]])
P += plot(lambda x: Li(x) - .5*Li(sqrt(x)),n-1000,n, color='red', legend_label=r'$Li(x)-\frac{1}{2}Li(\sqrt{x})$')
P += plot(F,n-1000,n, color='blue', legend_label=r'$\sum_{j=1}^{%s} \frac{\mu(j)}{j} Li(x^{1/j})$'%k)
show(P)
Where does the Moebius
in that approximation come from anyway?-
Since this wasn't enough, what else is involved in the error
Are there connections with things other than just
What does this have to do with winning a million dollars?