VMO 1972 Problem 2

For any positive integer n, let f(n) = ∑ (-1)$^{(d-1)/2}$ where the sum is taken over all odd d dividing n. Show that: f($2^n$) = 1 f(p) = 2 for p a prime congruent to 1 mod 4 f(p) = 0 for p a prime congruent to 3 mod 4 f($p^n$) = n+1 for p a prime congruent to 1 mod 4 f($p^n$) = 1 for p a prime congruent to 3 mod 4, and n even f($p^n$) = 0 for p a prime congruent to 3 mod 4, and n odd Show that f(mn) = f(m) f(n) for m and n relatively prime. Find f($5^411^{28}17^{19}$) and f(1980). Show how to calculate f(n).