REMARKS OF CONGRUENT ARITHMETIC SUMS OF THETA FUNCTIONS DERIVED FROM DIVISOR FUNCTIONS

In this paper, we study a distinction the two generating functions : ${\varphi}^k(q)=\sum_{n=0}^{\infty}r_k(n)q^n$ and ${\varphi}^{*,k}(q)={\varphi}^k(q)-{\varphi}^k(q^2)$ ($k$ = 2, 4, 6, 8, 10, 12, 16), where $r_k(n)$ is the number of representations of $n$ as the sum of $k$ squares. We also obtain some congruences of representation numbers and divisor function.