Let $\Lambda(n)$ denote the von Mangoldt function: $\Lambda(n)=\log p$ when $n=p^e$ is a prime power ($e\ge 1$) and $\Lambda(n)=0$ otherwise. and $\lambda(n)$ be Liouville Function, , I'm interested to know about average growth of the following function $T_j(x)=\sum_{n\leq x}\frac{\lambda(n)\Lambda(n)}{n^j} $ such that I want to determine its upper bound for large $n$ , Probably it is known that positivity of $T(n)=\sum_{k=1}^{n}\frac{\lambda(k)}{k} $ for large $n$ is an open problem such that its confirmation led to proof of the Riemann hypothesis according to Pál Turán ,My two dependents questions are :
Question:
a) Is $T(n)=\sum_{k=1}^{n}\frac{\lambda(k)\Lambda(k)}{k} \geq 0$ (Does this sum also behave like related sum to Pólya conjecture )
b) What is the upper bound of $T_j(x)=\sum_{n\leq x}\frac{\lambda(n)\Lambda(n)}{n^j} $ , take the case $j=0$?
Note: The motivation of this question is to look if the positivity of the titled sum also led to proof of Riemann hypothesis
