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?

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    $\begingroup$ $d(n)$ varies between $2$ and many. Severin Wigert proved $\limsup_{n\to\infty}\dfrac{\log d(n)}{\log n/\log\log n}=\log2$ $\endgroup$ – Henry May 9 at 9:04
  • $\begingroup$ @Henry thank you I'll look it up en.wikipedia.org/wiki/Carl_Severin_Wigert $\endgroup$ – onepound May 9 at 9:06

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