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

  • $\begingroup$ Your try on $\Lambda\ast\Lambda$ is correct (fix the LHS). For the $m$-fold Dirichlet convolution of $\Lambda$ with itself - call it $\Lambda_m$ - a formula like yours can be derived; another way to see it is from $\sum_{n>0}\Lambda_m(n)/n^s=(-\zeta'(s)/\zeta(s))^m$. Your last question is somewhat confusing ("elementary" but "not usually in the books"). Start with something like this and follow the links - what else could be recommended?.. $\endgroup$ – metamorphy Dec 23 '18 at 20:18
  • $\begingroup$ (I should have added "... and the expression for $-\zeta'(s)/\zeta(s)$ from the Euler product".) $\endgroup$ – metamorphy Dec 23 '18 at 20:29

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