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# Approximation for the $$n^\text{th}$$ prime number

The Prime Number Theorem states that $$\displaystyle \pi(x) \sim \frac{x}{\log{x}}$$, where $$\displaystyle \pi(x)$$ is the number of primes not exceeding $$\displaystyle x$$, that is, $$\displaystyle \lim _{x\to \infty }{\frac {\;\pi (x)\;}{\frac {x}{\log(x)}}}=1$$. Let $$\displaystyle p_{n}$$ be the $$\displaystyle n$$th prime number. Prove that, as a corollary of the Prime Number Theorem, $$\displaystyle p_{n} \sim n\log{n}$$, that is, $$\displaystyle p_{n} \to n\log{n}$$ as $$\displaystyle n \to \infty$$.

Note by Alexander Israel Flores Gutiérrez
2 months, 2 weeks ago