On the Rankin-Sobolev problem regarding extrema of Epstein's zeta- function. Estimate of the origin of the ray of extremality of Voronoi's second perfect form (Q2640636)

Let f denote a positive definite quadratic form in n variables, and define its Epstein zeta-function for Re s\(>1\) by \(\zeta (f;s)=\sum f(k)^{-ns/2}\), the sum being taken over \(k\in {\mathbb{Z}}^ n\setminus \{(0,...,0)\}\). If there are real numbers \(s_ 0\geq 1\) such that, for all \(s>s_ 0\), f is a local minimum point of \(\zeta\) (f;s) among the forms f with fixed determinant, let S(f) denote the smallest such \(s_ 0\). It is proved that for the perfect form \(\phi_ 1^{(n)}(x)=\sum_{1\leq i<j\leq n}x_ ix_ j-x_ 1x_ 2\) we have \(S(\phi_ 1^{(n)})<2\) for every \(n\geq 5\).