The asymptotic of the first Chebyshev function, using the Prime Number Theorem

Question

Using the prime number theorem, show that:

$\vartheta (x) \sim x$

Where $\vartheta (x) := \sum_{p \le x} \log p$

Any help on this would be great, thanks in advance.

Answer 1

Just notice that $$\vartheta (x)=\int_1^x \ln t\ \text{d}\pi(t)=\pi(x)\ln x-\int_1^x\dfrac{\pi(t)}{t}\ \text{d}t$$ Applying Prime number theorem $$\pi(x)\sim \dfrac{x}{\ln x}$$ Q.E.D.