The first two parts look good. For the last part, I think at first, consider p as a prime.
What is in this case?
Problem:
Let f be the arithmetic function given by f(n) = n if n is square-free and f(n) = 0 if n is not square-free.
(a) Prove that f is a multiplicative function.
(b) Is f completely multiplicative?
(c) Let F be the summatory function of f: . Find for any a. Calculate F(144).
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Attempt:
A square-free number is a number that its unique prime factors do not repeat. 4(which is 2*2), 8(which is 2*2*2), 9(which 3*3).. are not square-free.
So, for part (a), I would like to show for some m, n, then
f(mn) = f(m)f(n) by the definition of multiplicative.
In order to show this, I broke it into cases.
If m, n are not square free then f(mn) = 0 = f(m)(n)
If either m or n are not square free then f(mn) = 0, so either f(m) = 0 or f(n) = 0.
If both m and n are square free and , then f(mn) = m*n=f(m)f(n)
(b) I say this is not completely multiplicative since if m = n, then f(mn) = f(mm) = f(nn) = not a square free number; thus, f(mn) = 0.
(c) , but I don't know how to derive a formula for the summatory function F(n). I know that since small f is multiplicative, then so is big F.