QUADRATIC FORMS REPRESENTING ALL ODD POSITIVE INTEGERS

• Published in: American journal of mathematics 2014-12, Vol.136 (6), p.1693-1745
• Main Author: Rouse, Jeremy
• Format: Article
• Language: English
• Subjects: Coefficients; Conceptual lattices; Discriminants; Escalators; Fourier coefficients; Inner products; Integers; Linear equations; Mathematical lattices; Mathematical problems; Numbers; Odd numbers; Quadratic programming; Theorems
• ISSN: 0002-9327; 1080-6377; 1080-6377
• DOI: 10.1353/ajm.2014.0041

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 $\mathrm{Q}\left(\overrightarrow{\mathrm{x}}\right)=\mathrm{n}$ has no integer solutions is O(D2+∈).