I'm self-studying analytic number theory in tao's blog. There is an exercise that I can't solve.
Let ${k \geq 1}$. Show that ${\Lambda_{2k} + \Lambda_k * \Lambda_k}$ can be expressed as a linear combination of convolutions of the form ${\mu * \dots * \mu * L^{a_1} * \dots * L^{a_r}}$, where ${\mu}$ appears ${k}$ times and ${a_1,\dots,a_r}$ are non-negative integers with ${a_1+\dots+a_r = 2k}$ and ${r \leq k}$. Identities of this form are due to Diamond and Steinig.
Here $L=\log $ is the standard natural logarithm restrited to the natural numbers and $\Lambda_k$ is the higher-order von Mangoldt functions defined by $\Lambda_k:=\mu*L^k$.
I think an induction may work. Let's first prove the base case $k=1$, then $$\Lambda_{2k}+\Lambda_k*\Lambda_k=\Lambda_2+\Lambda*\Lambda=\mu*L^2+(\mu*L)*(\mu*L),$$ I'm stuck here, I don't know how to simplify the expression.