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Facts on Prime Counting Function, True/False

NUMTH-12JXSS

Let $x\geq 1$ be a real number. Denote by $\pi(x)$ the number of prime numbers up to $x$, so we write

$$ \pi(x):= \sum_{p\leq x} 1.$$

Which of the following is true? Select ALL that apply.

A

$\pi(x)\leq x$.

B

$\lim\limits_{x\rightarrow\infty} \pi(x)=\infty$.

C

$\lim\limits_{x\rightarrow\infty} \frac{\pi(x)\log x}{x}=1$.

D

$\lim\limits_{x\rightarrow\infty} \frac{\pi(x)}x=1$.

E

$\pi(2x)-\pi(x)\geq 1$.