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

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

The Prime Number Theorem (PNT) states that

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

Denote by $\theta(x)$ and $\psi(x)$ the Chebyshev's functions defined by:

$$ \theta(x) = \sum_{p\leq x} \log p, \ \ \psi(x)= \sum_{n\leq x} \Lambda(n) $$

...where $\Lambda(n)$ is the von-Mangoldt function.

Denote by $\zeta(s)$ the Riemann zeta function. Let $p_n$ be the $n$-th prime number in increasing order.

Which of the following statements are equivalent to the PNT? Select ALL that apply.


The Riemann Hypothesis: All nontrivial zeros of $\zeta(s)$ satisfy $\mathrm{Re}(s)=\frac12$.


$\theta(x)\sim x$.


$\psi(x)\sim x\log x$.


There are no zeros of $\zeta(s)$ satisfying $\mathrm{Re}(s)=1$.


$p_n\sim n\log^2 n$.

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