Ramanujan's ternary quadratic form
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 A correct version of all details for the case of the quadratic form \(x^2+y^2+18 z^2\) is given in the unpublished diploma thesis by \textit{Thomas Reinke} (Münster 1999).
scientific article; zbMATH DE number 1146055