# Can the prime number theorem be given in terms of d(x) or σ(x)?

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?

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