Is $e^{\frac{\vartheta(x)}{x}}\ll_{\varepsilon}x^{\varepsilon} $?

Question

The question is in the title : let $ \vartheta(x) : =\displaystyle{\sum_{p\leqslant x}\log p} $ denote the first Chebyshev function.

Is it true that $ \forall\varepsilon>0,e^{\frac{\vartheta(x)}{x}}\ll_{\varepsilon}x^{\varepsilon} $?

Answer 1

The prime number theorem is equivalent to $\vartheta(x)/x \to 1$. Hence $\vartheta(x)/x \ll \varepsilon \log x$.