I am sorry for being rude in my previous thread, I would like to apologize and resubmit my question with all the correct notation and hopefully present it more clearer. My text is Elementary Number Theory by Jones and Jones. I do not understand Dirichlet product and series well at all. I am sure the definitions and lemmas in my text would be enough to tackle these problems, but I cannot seem to grasp them. Any help would be greatly appreciated, and I thank you for your time.

$\displaystyle k$ is a positive integer and $\displaystyle f,g$ and $\displaystyle h$ are arithmetic functions. $\displaystyle \hat{f}$ denotes the Dirichlet series for $\displaystyle f$, so

$\displaystyle \hat{f}(s)$ = $\displaystyle \sum_{n=1}^{\infty}{f(n)/n^s}$

Suppose $\displaystyle h$ is completely multiplicative. Define $\displaystyle f^{*k}$ inductively by $\displaystyle f^{*1}$=$\displaystyle f$and $\displaystyle f^{*(k+1)}$=$\displaystyle f*f^{*k}$.

Prove that:

(a)$\displaystyle (fh)*(gh) = (f*g)h$

(b)$\displaystyle (fh)*(gh)^{*k} = (f*g^{*k})h$

(c)$\displaystyle (fh)*h = (f*u)h$

(d)$\displaystyle (fh)*h^{*k} = (f*u^{*k})h$

(e)$\displaystyle h^{*k} = u^{*k}h$

(f) If $\displaystyle h$ is non-zero, then $\displaystyle (\mu^{*k}h) * h^{*k} =I$