Congruences for rs(n)

Let rs(n) denote the number of representations of n as the sum of s squares of integers. In this note, we determine rk2s(n) modulo 2k+2 for k⩾1 and s odd. For general s, we also obtain a congruence for rs(n) modulo 2s. These extend Wagstaff's results (J. Number Theory 127 (2007) 326–329).