On the Representation of Integers as Sums of an Even Number of Squares or Of Triangular Numbers

(1.4) Q(n) ns r2s(n) __ (-))! (e) nO( s -> 3, M It=1 ~~(s-i!m where e is any positive integer and se (s) is 1, 0 or 2s according as Q is odd, e 2 mod 4 or e = 0 mod 4. For --1 this says that the function nl-sr28(n) is in the mean (on the average) equal to ITS/(s 1)! if s > 3. For other values of e the formula implies a weighted asymptotic average of the same function, the weight factors being re(n) for varying n. These are therefore suitable functions to smooth out on the average the irregularities of the function n1ls r2S(n). What is proved is that the irregularity is smoothed out in the limit as mn becomes infinite; some empirical evidence (not recorded in the paper) indicates that the smoothing effect probably is generally in evidence for rather small values of m. The paper also contains results similar to (1.4) concerning representations of integers as sums of an even number of triangular numbers and also