Tying some pieces regarding the Zeta Function and the Prime Number Theorem together I came across two remarks that I would appreciate help in making the connections.
I) In Riemann's Explicit Formula: for $x > 1$
$\Pi = Li(x) - \sum_{\rho:\zeta(\rho)=0}Li (x^{\rho})- \log(2) +$ term relating to trivial zeros of $\zeta$ 
the $Li (x)$ term comes from the pole of $\zeta(s)$ at $s = 1$.
I know there is a pole at $s = 1$ (a.k.a. the harmonic series), but how does one show that the $Li(x)$ comes from this?
II) I understand the proof showing the non-trivial zeros $\rho$ of $\zeta(s)$ satisfy
$0 < Re(\rho)< 1$.
How is this equivalent to the Prime Number Theorem?
(The essence of the questions come from remarks in Stopple's "Primer of Analytic Number Theory.") 
I have a feeling that these questions are asking a lot, so I appreciate any help.
$EDIT$: The link provided in the comment below by Raymond Manzoni provides the answer to part I.
 A: To see why the Prime Number theorem is implied, by definition,
$$\Pi(z)=\sum_{r=1}^\infty\frac{1}{r}\pi(z^{1/r}).$$
Instead of $\Pi(x)$, consider Von Mangoldt's formula, using Chebyshev's prime counting function:
$$\psi(z):=\sum_{p \ \mbox{prime}, \ p^k\leq z}\ln p=\sum_{n\leq z}\Lambda(n),$$
where the sum is over all prime powers not exceeding $z$. 
Then the $\Pi$ formula above is equivalent to:
$$\psi(z)=z-\ln(2\pi)-\frac{1}{2}\ln(1-z^{-2})-\sum_{\rho}\frac{z^\rho}{\rho}$$
where $\rho$ are the nontrivial zeros of the zeta function. The prime number theorem is equivalent to showing that $\psi(z)\approx  z$. Write $\rho=a+i\sigma$. So $|z^\rho|=|z|^a$. Now try to show that if $0<a<1$, then the sum is of lower order than $z$. Making this argument rigorous is nontrivial but I believe it is in spirit with de la Valee Poussin's original proof of the prime number theorem.
A: Concerning question I the explicit formulas link (I updated the $(2)-(4)$ part for clarification) should help to find the origin of the $\operatorname{li}(x^{\rho})$ for the Riemann explicit formula starting from the Euler product formula ($\zeta$ as function of the primes) up to the explicit formula for $\pi(x)$ (the primes as function of the $\zeta$ zeros).
A detailed exposition of von Mangoldt's proof for this formula is in pages $62$ to $65$ of Edwards' hard to replace book 'Riemann's Zeta Function'.

(Notation: in ANT it is usual to note '$\sigma$' the real part of the complex number $\,s:=\sigma+it\,$)
Concerning question II I'll reproduce verbatim Ingham's demonstration ( page $37$) that
$\qquad\qquad$PNT $\implies$ {no zeros on the line $\sigma=1$}
Let's start with the equation $(2.1)$ from the explicit formulas link :
$$\tag{1}f(s):=-\frac{\zeta'(s)}{\zeta(s)}=s\int_1^{\infty}\frac{\psi(x)}{x^{s+1}}dx\quad (s>1)$$
"we have, for $\,\sigma>1$,
$$\tag{2}\phi(s):=\int_1^{\infty}\frac{\psi(x)-x}{x^{s+1}}dx=-\frac{\zeta'(s)}{s\;\zeta(s)}-\frac 1{s-1}$$
say; $\phi(s)$ is regular in $\sigma>0$ except (possibly) for simple poles at zeros of $\zeta(s)$. Now suppose the PNT true, i.e. $\psi(x)=x+o(x)$. Then, given $\epsilon>0$, we have $\,|\psi(x)-x|<\epsilon x\;$ for $\;x>x_0=x_0(\epsilon)\;(>1)$. Hence, for $\sigma>1$,
$$|\phi(s)|<\int_1^{x_0}\frac{|\psi(x)-x|}{x^2}dx+\int_{x_0}^\infty\frac{\epsilon}{x^\sigma}dx<K+\frac{\epsilon}{\sigma-1},$$
where $K=K(x_0)=K(\epsilon)$. Thus
$$|(\sigma-1)\phi(\sigma+ti)|<K(\sigma-1)+\epsilon<2\epsilon$$
for $\,1<\sigma<\sigma_0=\sigma_0(\epsilon,K)=\sigma_0(\epsilon)$. Hence, for any fixed $t$,
$$\tag{3}(\sigma-1)\,\phi(\sigma+ti)\to 0$$
as $\sigma\to 1+0$. This shows that the point $1+ti\,$ cannot be a zero of $\zeta(s)$, for in that case $(\sigma-1)\phi(\sigma+ti)$ would tend to a limit different from $0$, namely the residue of $\phi(s)$ at the simple pole $1+ti$."

The converse implication : $\qquad${no zeros on the line $\sigma=1$} $\implies$ PNT
is not so direct since the usual proofs require {no zeros on the line $\sigma=1$} but also a theorem on the order of magnitude of $\frac{\zeta'(s)}{\zeta(s)}$ to imply the PNT (c.f. the discussion page $37-39$ of Ingham).
This subsidiary theorem is something like Hardy & Littlewood's $f(\sigma+it)=O(|t|^\alpha)$ with $\alpha<1$, for $\sigma\ge 1$ and large $|t|$.
In fact $\zeta(s)=O(\ln|t|)$ and $\zeta'(s)=O\bigl(\ln^2|t|\bigr)$) were obtained as well as $\frac{\zeta'(s)}{\zeta(s)}=O\bigl(\ln^9|t|\bigr)$ for $\rho>1-A\,\ln^{-9}|t|$ allowing to find a large 'zero-free region' along $\sigma=1$. I think that this additional requirement came from the infinite bounds of : 
$$\tag{4}\psi^*(x)=\frac1{2\pi i}\int_{c-i\infty}^{c+i\infty}f(s)\frac{x^s}s\,ds$$
This initial point of view (de la Vallée-Poussin and Hadamard's) is well exposed in Titchmarsh reference book 'The theory of the Riemann Zeta-function' (around page $50$). 
$$-$$
An important progress was made when Wiener, combining his work about Fourier transforms and Lambert series with Ikehara's Theorem, obtained the Tauberian theorems he exposed in two books : 1932 : 'Tauberian theorems' (a $100$ pages paper accessible after free registration at JStor) and 1933 : 'The Fourier integral and certain of its applications' (ch.$19$ 'Ikehara's Theorem').
The 'Wiener–Ikehara theorem' theorem asserts (Chandrasekharan) :
If $A(t)$ is a non-negative, non-decreasing function of $t$, defined for $t\ge 0$ and if the integral
$$\int_0^\infty A(t)\,e^{-ts}\,dt$$
is convergent for $\sigma>1$ to the function $f(s)$ analytic for $\sigma\ge 1$ except for a simple pole at $s=1$ with residue $1\,$ then :
$$\lim_{t\to\infty} \ e^{-t}\,A(t)=1$$
(for a proof and many more informations concerning the PNT see Montgomery and Vaughan's book on 'Multiplicative NT' page $259$ or Chandrasekharan's 'Introduction to ANT' p.$124$ or Wiener's work)
After setting $x:=e^t$ in equation $(1)$ we get (for $\sigma>1$) :
$$-\frac{\zeta'(s)}{s\;\zeta(s)}=\int_0^{\infty}\psi(e^t)\,e^{-ts}\,dt$$
Let's apply the WIT to the Chebyshev function $\;A(x):=\psi\bigl(e^x\bigr)$ then :


*

*$\psi$ is non-decreasing and $\psi\bigl(e^x\bigr)\ge 0$

*$\zeta(s)$ and $\zeta'(s)$ are analytic for $\sigma>0$ except at $s=1$ where $\,\frac{\zeta'(s)}{s\;\zeta(s)}$ admits a simple pole

*$\zeta(s)$ does not vanish in the half-plane $\sigma\ge 1$ (this is where the hypothesis $\zeta(s)\not = 0$ for $\sigma=1$ appears since the other cases are well known)


All this implies that $\,\psi\bigl(e^t\bigr)\sim e^t\;$ or $\;\psi(x)\sim x\;$ as $\;x\to \infty$ (i.e. the PNT). Of course all the 'machinery' is in the WIT here ! ( or the reverse ? ;-) )
and we got the wished direct implication : $\qquad${no zeros on the line $\sigma=1$} $\implies$ PNT
adding only continuity in the close half-plane $\sigma\ge1$ of $\;\displaystyle \zeta(s)-\frac 1{s-1}$ and $\;\displaystyle\zeta'(s)+\frac 1{(s-1)^2}$.
$$-$$
In $1980$ Newman proposed another proof of the PNT using (in Korevaar's words) a 'poor man' version of Wiener–Ikehara's. His proof required only the analyticity and non-vanishing of $(s-1)\zeta(s)$ on the closed half-plane $\{s:\Re(s)\ge1\}$) (i.e. well known facts except on the $\Re(s)=1$ line).  
Newman's theorem may be rewritten with the Laplace integral replacing the Dirichlet series (Korevaar and Zagier) :
Let $A(t)\;$ be a bounded on $(0,\infty)$ and locally integrable function and suppose that the function $$g(s):=\int_0^\infty A(t)\,e^{-ts}\,dt,\quad \Re(s)>0$$ extends holomorphically to $\,\Re(s)\ge 0$ then the limit as $s\to 0$ exists and
$$\int_0^\infty A(t)\,dt=g(0)$$
Newman proposed different proofs of the PNT. The more direct is to use the formula from inversion of Dirichlet series for $\Re(s)>1$ : $\,\displaystyle\frac 1{\zeta(s)}=\sum \frac{\mu(n)}{n^s}$. Since further $(s-1)\zeta(s)$ is analytic and zero free over $\Re(s)\ge 1$ the theorem (in its Dirichlet form) applies and we get convergence to $\,\displaystyle\sum \frac{\mu(n)}{n}=0$ which, according to Landau is equivalent to (for Hardy 'as deep as') the prime number theorem. 
For his proof Newman modified a theorem of Ingham (using Fourier analysis) and came back to contour integral with the idea of replacing $(4)$ by a finite $C_R$ contour integral. The Cauchy formula gave directly (concluding with the limit as $R\to\infty\;$ that $\;\lim_{T\to\infty} \{\text{left part}\}=0$) :
$$g(0)-g_T(0)=\frac1{2\pi i}\int_{C_R}\bigl(g(s)-g_T(s)\bigr)\,x^s\left(\frac 1s+\frac s{R^2}\right)\,ds,\quad g_T(s):=\int_0^T A(t)\,e^{-ts}\,dt$$
(the complete and very short proof of Zagier should be examined !)
This work is described in a few nice papers :


*

*Newman in Simple analytic proof of the prime number theorem (and his ANT book)

*Korevaar in On Newman's Quick Way to the Prime Number Theorem and 

*Zagier in Newman's Short Proof of the Prime Number Theorem for a very short proof. 


You may enjoy these last references as well as the history of all this exposed by Bateman and Diamond in 'A hundred years of Prime Numbers'.
