Quadratic forms representing all odd positive integers

TL;DR

This paper classifies positive-definite quadratic forms that represent all positive odd integers, proving a finite check suffices and introducing a new analytic method based on Rankin-Selberg L-functions.

Contribution

It establishes a finite criterion for representing all positive odd integers and develops a novel analytic approach to bound cusp constants of quadratic forms.

Findings

Proves that representing all odd integers up to 451 suffices for certain forms.

Introduces a new method to bound cusp constants using Rankin-Selberg L-functions.

Shows that three remaining ternary forms represent all positive odd integers assuming GRH.

Abstract

We consider the problem of classifying all positive-definite integer-valued quadratic forms that represent all positive odd integers. Kaplansky considered this problem for ternary forms, giving a list of 23 candidates, and proving that 19 of those represent all positive odds. (Jagy later dealt with a 20th candidate.) Assuming that the remaining three forms represent all positive odds, we prove that an arbitrary, positive-definite quadratic form represents all positive odds if and only if it represents the odd numbers from 1 up to 451. This result is analogous to Bhargava and Hanke's celebrated 290-theorem. In addition, we prove that these three remaining ternaries represent all positive odd integers, assuming the generalized Riemann hypothesis. This result is made possible by a new analytic method for bounding the cusp constants of integer-valued quaternary quadratic forms $Q$ with fundamental discriminant. This method is based on the analytic properties of Rankin-Selberg $L$-functions, and we use it to prove that if $Q$ is a quaternary form with fundamental discriminant, the largest locally represented integer $n$ for which $Q(\vec{x}) = n$ has no integer solutions is $O(D^{2 + \epsilon})$.