Remarks of congruent arithmetic sums of theta functions derived from divisor functions

Abstract

In this paper, we study a distinction the two generating functions : ϕ k (q) = P∞ n=0 rk(n)q n and ϕ ∗,k(q) = ϕ k (q) − ϕ k (q 2 ) (k = 2, 4, 6, 8, 10, 12, 16), where rk(n) is the number of representations of n as the sum of k squares. We also obtain some congruences of representation numbers and divisor function.