P3. (3+5 points) Let k be any natural number. (a) Prove that for every positive integer n, we have σk(n) = σ−k(n)n k . Conclude
that n is a perfect number exactly when σ−1(n) = 2. (b) Prove that for all positive integers n, we have σ1(n) ≤ n log(n + 1) + γn, where γ is Euler’s constant defined in class. (In this course, log x = loge x denotes the natural logarithm). g
