Analytic Number Theory Question:

Please do not answer unless you are sure and please write clearly.

Consider a function f, we definef~(n)=1/d(n)mnf(m) as the average value of f on the divisors of n.

Prove the Following Statements:

1.) Prove that if f is multiplicative, so isf~

2.) Let f(n)=2n. Show thatf~ =d(n2)/d(n)

3.) Let f(n) = 2n. Show thatf(n)~=pan2a+11/a+1

4.) iff=f~ then f is constant (i.e. f(n)= 1 for allnN