On representation of an integer as a sum by X^2+Y^2+Z^2 and the modular equations of degree 3 and 5

Alexander Berkovich

Published 2009-07-10, updated 2012-07-04Version 4

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 and x^2+y^2+3z^2+xy, respectively. I propose an interesting identity for s(p^2n)- p s(n) with p being an odd prime. This identity makes nontrivial use of the ternary quadratic forms with discriminants p^2, 16p^2.