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$P(x) = \sum_{k=1}^{\infty} \frac 1k \pi(x^{1/k})$

and

$Li(x) = \int_2^n \frac {dt}{\log t}$

And the prime number theorem is:

$$\pi(n) \sim \frac{n}{\log n }$$

I want to show that $$P(x) \sim Li(x)$$ is equivalent to prime number theorem.

Can some body please help me with this.

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# Prime number theorem

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