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