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Universal sums via products of Ramanujan's theta functions

Nasser Abdo Saeed Bulkhali, Zhi-Wei Sun

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.

Universal sums via products of Ramanujan's theta functions

Abstract

An integer-valued polynomial is said to be universal (over ) if each nonnegative integer can be written as with . In this paper, we mainly introduce a new technique to determine the universality of some sums in the form (with all even) conjectured by Sun, using various identities of Ramanujan's theta functions. For example, we prove that and are universal.

Paper Structure

This paper contains 5 sections, 14 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Universal sums via the theory of quadratic forms
  3. An approach via Ramanujan's theta functions
  4. Proof of Theorem \ref{['Th1.2']}
  5. Proof of Theorem \ref{['Th1.4']}

Key Result

Theorem 1.1

Let $i,j,s,t,u,v\in\Bbb Z$ with $i+j,s+t,u+v$ positive. Suppose that where $a_{1r},a_{2r},b_{1r},b_{2r},c_{1r},c_{2r}\in\Bbb Z$ and $k,m_r,a_{1r}+a_{2r},b_{1r}+b_{2r},c_{1r}+c_{2r}\in\Bbb Z^+$. (i) The tuple $(i+j,i-j,s+t,s-t,u+v,u-v)$ is universal if and only if is universal for all $r=1,\ldots,k$. (ii) The tuple $(i+j,i-j,s+t,s-t,u+v,u-v)$ is almost universal if and only if the tuple is almos

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Theorem 1.3
  • Lemma 2.1
  • Remark 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 2.5
  • ...and 7 more