I am currently trying to understand a proof of the prime number theorem of Terence Tao. I think I understand nearly everything but the proof of the implication of the theorem, that $\zeta(s)$ has no zeros on the line {$1+it:t\in\mathbb{R}$}, to the von Mangoldt form of the prime number theorem: $\lim_{x\rightarrow\infty}\frac{1}{x}\sum_{n\leq x}\Lambda(n)=1$. I will try to describe the proof of the implication as short and as detailed as I can but if you're interested in the whole proof of Terence Tao, then here is the link (You need to scroll down a fair bit to get to the point I'm talking about): https://terrytao.wordpress.com/tag/prime-number-theorem/
The main tool used for the proof is an exercise he gave:
It holds that \begin{equation} \sum_{n}\Lambda(n)\eta(n)=\int_{R}\left(1-\frac{1}{y^3-y}\right)\eta(y)\text{dy}-\sum_{\rho}\int_{R}\eta(y)y^{\rho-1}\text{dy} \end{equation} whenever $\eta:\mathbb{R}\rightarrow\mathbb{C}$ is a smooth function, compactly supported in $(1,+\infty)$ with the summation being absolutely convergent. These $\rho$ are the complex zeros of the $\zeta$-function.
Now he constructed such a function $\eta_{T,x}$, with $2\leq T\leq x$ , that equals one on $\lbrack 2,x\rbrack$ and is supported on $\lbrack1.5,x+\frac{x}{T}\rbrack$. Furthermore the function obeys the following derivative estimates \begin{equation} \eta^{(j)}(y)\leq C_j\text{ for all }y\in\lbrack 1.5,2\rbrack,j\geq 0,C_j\in\mathbb{R} \end{equation} and \begin{equation} \eta^{(j)}(y)\leq\left(\frac{T}{x}\right)^{j}C_{j}\text{ for all }y\in\lbrack x,x+T\rbrack,j\geq 0,C_{j}\in\mathbb{R}. \end{equation}
As far as I understand, from the given exercise follows \begin{equation} \sum_{n\leq x}\Lambda(n)\leq\sum_{n}\Lambda(n)\eta(n)=\int_{\mathbb{R}}\eta(y)\text{dy}-\int_{\mathbb{R}}\frac{\eta(y)}{y^3-y}\text{dy}-\sum_{\rho}\int_{\mathbb{R}}\eta(y)y^{\rho-1}\text{dy}. \end{equation} Here he needed to find estimates of these three integrals. Thats the part that I don't understand. He wrote without further explenation, that \begin{equation} \int_{\mathbb{R}}\eta(y)\text{dy}=x+O(\frac{x}{T}) \end{equation} and \begin{equation} \int_{\mathbb{R}}\frac{\eta(y)}{y^3-y}\text{dy}=O(1). \end{equation}
I tried to plug in the function $\eta(y)$ with respect to its definition, but that's not taking me anywhere. Maybe I don't have a good enough grasp of this $\eta$-function, but I do not understand where these estimates are coming from. Anyway, he concludes that \begin{equation} \sum_{n\leq x}\Lambda(n)\leq x + O(\frac{x}{T})+\sum_{\rho}O\left(\left|\int_{R}\eta(y)y^{\rho-1}\text{dy}\right|\right). \end{equation} For the last integral, he looks at the cases $|\rho|\leq T_{\*}$ and $|\rho| > T_{\*}$, where $T_{\*}\geq T$.
For $|\rho|\leq T_{\*}$ there are only $O_T(1)$ zeros and because we say that there is no zero on the line {$1+it:t\in\mathbb{R}$} all of them have real part strictly less than $1$. Hence, there exists $\epsilon=\epsilon_{T_{\*}}>0$, such that $\Re(\rho)\leq 1-\epsilon$.
Again he concludes, only saying that he used the triangle inequality, that \begin{equation} \int_{\mathbb{R}}\eta(y)y^{\rho-1}\text{dy}\leq x^{1-\epsilon}C_{T_{\*}}. \end{equation} For the other case, he partially integrated two times and concluded with $\Re(\rho)\leq 1$ \begin{equation} \int_{\mathbb{R}}\eta(y)y^{\rho-1}\text{dy}=\frac{1}{\rho(\rho+1)}\int_{\mathbb{R}}\eta^{''}(y)y^{\rho+1}\text{dy}\leq \frac{1}{|\rho|^2}\frac{T^2}{x^2}\frac{x}{T}x^2=\frac{T}{|\rho|^2}x. \end{equation}
Again, I do not have any idea how to get these estimates. Any explanation would be greatly appreciated.
And thus he has \begin{equation} \sum_{n\leq x}\Lambda(n)\leq x + O(\frac{x}{T})+O_T(x^{1-\epsilon}) \end{equation} \begin{equation} \Leftrightarrow\sum_{n\leq x}\Lambda(n)\leq x+O(\frac{x}{T}). \end{equation} Sending $T\rightarrow\infty$ gives the upper bound and a similar argument gives the lower bound and thus the von mangoldt form of the prime number theorem.
Any help trying to understand these integrals and their estimates is greatly appreciated. Many thanks in advance.