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# State/Chebyshev estimates for the prime number function

Theorem: (Chebyshev estimates for the prime number function) For all $\epsilon > 0$, there exists $x_0 > 0$, such that for all $x \in \NN$ satisfying $x > x_0$, $$(\log(2) - \epsilon) \frac{x}{log(x)} \leq \pi(x) < (\log(4) + \epsilon) \frac{x}{\log(x)},$$ where $\pi(x)$ denotes the prime counting function, and $log(x)$ denotes the natural logarithm of $x$.

## Logical Connections

This statement logically relies on the following definitions and statements: Def/Prime counting function, Def/Natural logarithm, State/Estimate for the product of primes

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