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

Abstract

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 $$\\\\\\\\displaystyle{ s(25n) = \\\\\\\\left (6 -\\\\\\\\left (-n\\\\\\\\vert 5ight)ight)s(n) - 5s\\\\\\\\left (\\\\\\\\frac{n} {25}ight) }$$ 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 $$\\\\\\\\displaystyle{2{x}^{2} + 2{y}^{2} + 2{z}^{2} - yz + zx + xy,\\\\\\\\quad {x}^{2} + {y}^{2} + 3{z}^{2} + xy,}$$ respectively. Finally, I propose a remarkable new identity for s ( p 2 n )鈭p s ( n ) with p being an odd prime. This identity makes nontrivial use of the ternary quadratic forms with discriminants p 2 , 16 p 2 .