Given according to Apostol's Introduction to Analytic Number Theorem p79, these relations satisfy equivalence to the Prime Number Theorem:
$\lim_{x\to \infty}\frac{\pi(x)\ln(x)}{x}=1$
$\lim_{x\to \infty}\frac{\theta(x)}{x}=1$
$\lim_{x\to \infty}\frac{\psi(x)}{x}=1$
They are in terms of $\pi(x),\,\, \theta(x)\,\, and\,\, \psi(x)$, so can the same be done in terms of the divisor functions $d(x)$ or $\sigma(x)$ for the prime number theorem?