Zeta Equivalence to Prime Number Theorem

Theorem
Let $\zeta \left({z}\right)$ be the Riemann $\zeta$ function.

The Prime Number Theorem is logically equivalent to the statement that the average of the first $N$ coefficients of $\dfrac{\zeta'}{\zeta}$ tend to $-1$ as $N$ goes to infinity.

Proof
The Von Mangoldt Equivalence is equivalent (clearly) to the statement that the average of the coefficients of the function of $z$ defined as:
 * $(1): \quad \displaystyle \sum_{n \mathop = 1}^\infty \frac{\Lambda \left({n}\right)}{n^z}$

tend to $1$.

Let $ \left\{{ p_1, p_2, p_3, \dots }\right\}$ be an enumeration of the prime numbers:
 * $\left\{{ 2, 3, 5, 7, 11, \dots }\right\}$

In the proof of the Von Mangoldt Equivalence, in the sum of von Mangoldt function, the $\ln \left({p}\right)$ term will appear once for each power of $p$.

So, we expand out $(1)$ as:

This function of $z$ can be recognized as:

Hence the result.