Ramanujan's ternary quadratic form
type: Item
instance of: scholarly article
title: Ramanujan's ternary quadratic form
zbMATH Open document ID: 0930.11022
author: Ken Ono
author: Kannan Soundararajan
published in: Inventiones Mathematicae
publication date: 2000-02-14T00:00:00Z
review text: The ternary quadratic form \(x^2+y^2+10z^2\) represents all positive integers except those of type \(4^\lambda(16\mu+6)\) and finitely many exceptions 3, 7, 21, 31, 33, 43, 67, 79, 87, 133, 217, 219, 223, 253, 307, 391, 679, 2719, \dots. The authors conjecture that the 18 exceptions given above are all. They prove this conjecture under the assumption that the nontrivial zeros of all Dirichlet \(L\)-functions \(L(s,\chi)\) with a primitive real character \(\chi\) have real part \(\frac12\) and that the nontrivial zeros of the Hasse-Weil \(L\)-functions \(L(E(-10 N),s)\) (with \(N\) a square free integer coprime to 10) have real part 1. (Here \(E(D)\) is the elliptic curve given by \(y^2=x^3+Dx^2+4D^2x+4D^3)\). The proof uses Waldspurger's theorem connecting the Fourier coefficients of half-integer weight cusp forms with the central value of the \(L\)-function of their Shimura lift, furthermore technically complicated explicit estimates for Dirichlet \(L\)-functions in order to show that all exceptions (under the above assumptions) are \(<2\cdot 10^{10}\), the rest is done by computer. Lemma 5 is incorrect (noticed by Th. Reinke (Münster)). A correct version of all details for the case of the quadratic form \(x^2+y^2+18 z^2\) is given in the unpublished diploma thesis by \textit{Thomas Reinke} (Münster 1999).
Mathematics Subject Classification ID: 11E20
Mathematics Subject Classification ID: 11M06
zbMATH DE Number: 1146055
zbMATH Keywords: Dirichlet \(L\)-functions
zbMATH Keywords: ternary quadratic form
zbMATH Keywords: finitely many exceptions
Wikidata QID: Q55952541
reviewed by: Meinhard Peters
MaRDI profile type: Publication
full work available at URL: s002220050191
OpenAlex ID: W2056634454
DOI: 10.1007/S002220050191
description: scientific article; zbMATH DE number 1146055