Minimum of the theta-function of three variables and solution of the Rankin-Sobolev problem in a three-dimensional space
Let \(f\) be a positive definite quadratic form with real coefficients in \(n\) variables, and let \({\mathbf P}_d\) denote the space of such forms with fixed determinant \(d\). The Epstein zeta function is \(Z(f, s)= \sum_{{\mathbf x}\in \mathbb{Z}^n\backslash \{\text{\textbf{0}}\}} (f({\mathbf x}))^{- ns/2}\) for \(\text{Re } s> 1\), and defined by analytic continuation for other values of \(s\) (\(s= 1\) is a pole). The theta-function is \(\vartheta(f, \alpha)= \sum_{{\mathbf x}\in \mathbb{Z}^n}\exp(- 2\pi\alpha f({\mathbf x}))\) for any real \(\alpha> 0\). The Rankin-Sobolev problem is to find all local and absolute minima of \(Z(f, s)\) as a function of \(f\) on the space \({\mathbf P}_d\) for given \(s> 0\), \(s\neq 1\). The Montgomery problem is the same question for the function \(\vartheta(f, \alpha)\) for a given \(\alpha> 0\). Both problems have been solved earlier for \(n= 2\). The author announces a complete solution for \(n= 3\) by stating that the only minimum points are the ones corresponding to the equivalence class of a suitable multiple of the form \(x^2_1+ x^2_2+ x^2_3+ x_1 x_2+ x_2 x_3+ x_3 x_1\). Detailed proofs will appear elsewhere.
scientific article; zbMATH DE number 815610