On the extremality rays of finally-extremal forms in seven variables (Q1920854)

Let \(f\) denote a positive define quadratic form in \(n\) variables, and definite its Epstein zeta function for \(\text{Re} s>1\) by \(Z(f,s) = \sum f({\mathbf x})^{-ns/2}\), the sum being taken over \({\mathbf x} \in\mathbb{Z}^n \backslash \{{\mathbf o}\}\). If there are real numbers \(s_0\geq 1\) such that, for all \(s>s_0\), \(f\) is a local minimum point of \(Z(f,s)\) among the forms \(f\) with fixed determinant, let \(S(f)\) denote the smallest such \(s_0\). Take \(n=7\), and \[ \begin{aligned} \varphi^7_0 & = x^2_1 + \cdots + x^2_7+x_1x_2 + \cdots + x_6x_7, \\ \varphi^7_4 & = \varphi^7_0 - \textstyle {{1\over 2}} (x_1x_2 + x_3x_4+x_3x_5 + x_4x_5+x_6x_7), \\ \varphi^7_{16} & = \textstyle {{4\over 3}} (x^2_1+ \cdots + x^2_6-x_1x_2-x_2x_3-x_3x_4-x_4x_5-x_6) + x^2_7 - {4\over 3} x_2x_7. \end{aligned} \] The author proves that \[ S(\varphi^7_4) \leq 3.14,\;S (\varphi^7_{16})\leq 2.47. \]