Universal sums via products of Ramanujan's theta functions

TL;DR

This paper introduces a new technique using Ramanujan's theta function identities to determine the universality of certain quadratic sums, proving specific forms are universal over integers.

Contribution

A novel method leveraging Ramanujan's theta function identities to establish the universality of specific quadratic sums conjectured by Sun.

Findings

Proved that x(3x+1)+y(3y+2)+2z(3z+2) is universal.

Proved that x(4x+r)+y(3y+1)/2+z(7z+1)/2 is universal for r=1,3.

Developed a new technique for analyzing universality of quadratic forms.

Abstract

An integer-valued polynomial $P(x,y,z)$ is said to be universal (over $\mathbb Z$) if each nonnegative integer can be written as $P(x,y,z)$ with $x,y,z\in\mathbb Z$. In this paper, we mainly introduce a new technique to determine the universality of some sums in the form $x(a_1x+a_2)/2+y(b_1y+b_2)/2+z(c_1z+c_2)/2$ (with $a_1-a_2,b_1-b_2,c_1-c_2$ all even) conjectured by Sun, using various identities of Ramanujan's theta functions. For example, we prove that $x(3x+1)+y(3y+2)+2z(3z+2)$ and $x(4x+r)+y(3y+1)/2+z(7z+1)/2\ (r=1,3)$ are universal.