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Here you are the best formula to calculate Pi(x) by means of a 4-dimensional integral:

[tex]\pi(t)=\frac{1}{4\pi^2}\int_0^t\int_{c-i\infty}^{c+i\infty}\int_{d-i\infty}^{d+i\infty}\int_0^{\infty}dxdsdqdn\frac{n^{-s+2}x^{q-1}LnR(qn}}{R(4-s)}[/tex]

Where R(s is the Riemann Zeta function...

[tex]\pi(t)=\frac{1}{4\pi^2}\int_0^t\int_{c-i\infty}^{c+i\infty}\int_{d-i\infty}^{d+i\infty}\int_0^{\infty}dxdsdqdn\frac{n^{-s+2}x^{q-1}LnR(qn}}{R(4-s)}[/tex]

Where R(s is the Riemann Zeta function...

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