On Representation of an Integer by X 2 + Y 2 + Z 2 and the Modular Equations of Degree 3 and 5 (original) (raw)

I discuss a variety of results involving s(n), the number of representations of n as a sum of three squares. One of my objectives is to reveal numerous interesting connections between the properties of this function and certain modular equations of degree 3 and 5. In particular, I show that s(25n) = (6 − (−n|5)) s(n) − 5s " n 25 " follows easily from the well known Ramanujan modular equation of degree 5. Moreover, I establish new relations between s(n) and h(n), g(n), the number of representations of n by the ternary quadratic forms 2x 2 + 2y 2 + 2z 2 − yz + zx + xy, x 2 + y 2 + 3z 2 + xy, respectively. I also find generating function formulae for various subsequences of {s(n)}, for instance 6 ∞ Y j=1 (1 − q 2j) 2 (1 − q 10j)(1 + q −1+2j) 4 (1 + q −3+10j)(1 + q −7+10j) = ∞ X n=0 s(5n + 1)q n. I propose an interesting identity for s(p 2 n) − ps(n) with p being an odd prime.