CONVOLUTION SUMS ARISING FROM DIVISOR FUNCTIONS

Let sigma(s)(N) denote the sum of the sth powers of the positive divisors of a positive integer N and let sigma(s)(N) = Sigma(d vertical bar N)(-1)(d-1)d(s) with d, N, and s positive integers. Hahn [12] proved that 16 Sigma(k < N) (sigma) over tilde (k) (sigma) over tilde (3) (N - K) = -(sigma) over tilde (5) + 2(N -1) (sigma) over tilde (3) (N) + (sigma) over tilde (1) (N). In this paper, we give a generalization of Hahn's result. Furthermore, we find the formula Sigma(N-1)(k=1) (sigma) over tilde (1)(2(n-m) K)(sigma) over tilde (3) (2(n) N - 2(n) k) for m (0 <= m <= n).