TITLE: Tying some pieces regarding the Zeta Function and the Prime Number Theorem together
QUESTION [8 upvotes]: 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.

REPLY [4 votes]: 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'.