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.

is a positive integer and and are arithmetic functions. denotes the Dirichlet series for , so

=

Suppose is completely multiplicative. Define inductively by = and = .

Prove that:

(a)

(b)

(c)

(d)

(e)

(f) If is non-zero, then