Hello..that's my question today..why can't we obtain [tex] \pi(x) [/tex] by solving the integral equation obtained from Euler's product:(adsbygoogle = window.adsbygoogle || []).push({});

[tex] \frac{log \zeta(s)}{s}= \int_{2}^{\infty}dx \frac{ \pi(x)}{x(x^{s}-1)} [/tex] ?

- Of course we can't solve it "Analytically" (or perhaps yes, i will take a look to "Numerical Recipes"...:grumpy: ) but we could solve it Numerically using some quadrature method for the Integral equation..or introducing the term inside the Kernel:

[tex] \pi(s) = \int_{2}^{\infty} \pi(x) \delta (x-s) [/tex] so the integral becomes a "Fredholm Integral Equation of Second Kind".... I know that an algorithm (either numerical or similar) must exist to solve any Integral equation Numerically...why not for the Prime counting function?...

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# Integral involving Pi(x)

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