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

TL;DR

This paper explores the connections between the number of representations of integers as sums of three squares and modular equations of degrees 3 and 5, revealing new identities and relationships with quadratic forms.

Contribution

It introduces new relations between representation functions and modular equations, including identities involving primes and quadratic forms with specific discriminants.

Findings

s(25n) relation derived from Ramanujan's modular equation

New connections between s(n), h(n), and g(n) functions

Identity involving s(p^2n) and prime p

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 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.