# Some Trys related to Von Mangoldt function, What is $\Lambda * \Lambda$

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

• 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?.. – metamorphy Dec 23 '18 at 20:18
• (I should have added "... and the expression for $-\zeta'(s)/\zeta(s)$ from the Euler product".) – metamorphy Dec 23 '18 at 20:29