Given Mangoldt function: $\Lambda(x)=\begin{cases}\ln p,&\text{if $x=p^k$}\\0,&\text{otherwise}\end{cases}$
I wonder what about $\Lambda* \Lambda$ where $*$ denotes the dirichlet multiplication of two arithmetic function.
From my try I think that $\Lambda(n)=\Lambda(p_1^{\alpha_1}p_2^{\alpha_2}p_3^{\alpha_3} \dots p_k^{\alpha_k})=\sum_{d|{p_1^{\alpha_1}p_2^{\alpha_2}p_3^{\alpha_3} \dots p_k^{\alpha_k}}}\Lambda(d)\Lambda(\frac{n}{d})=\begin{cases} 0,&\text{if $k=0$}\\ (\alpha_1-1)(\log p_1)^2,&\text{if $k=1$}\\ 2(\log p_1)(\log p_2),&\text{if $k=2$}\\ 0,&\text{otherwise}\end{cases}$ Is it correct?
What about $(\Lambda)*(\Lambda)*(\Lambda) * \dots * (\Lambda)$
I just started Analytic Number Theory and I am following Tom Apostol
I wanted to know this because of curiousity, It looks like $\Lambda$ is necessary function in Analytic Number Theory.
If there is some use of studying $\Lambda* \Lambda$
Can you please give me some elementary properties of Mangoldt Function which are usually not written in the books