On the minima and convexity of Epstein zeta function

Let $Zn(s;a1,…,an)$ be the Epstein zeta function defined as the meromorphic continuation of the function $∑k∊Zn\{0}(∑i=1n[aiki]2)−s,Res>n/2$ to the complex plane. We show that for fixed $s≠n/2$⁠, the function $Zn(s;a1,…,an)$ as a function of $(a1,…,an)∊(R+)n$ with fixed $∏i=1nai$ has a unique minimum at the point $a1=⋯=an$⁠. When $∑i=1nci$ is fixed, the function $(c1,…,cn)↦Zn(s;ec1,…,ecn)$ can be shown to be a convex function of any $(n−1)$ of the variables ${c1,…,cn}$⁠. These results are then applied to the study of the sign of $Zn(s;a1,…,an)$ when $s$ is in the critical range $(0,n/2)$⁠. It is shown that when $1≤n≤9$⁠, $Zn(s;a1,…,an)$ as a function of $(a1,…,an)∊(R+)n$ can be both positive and negative for every $s∊(0,n/2)$⁠. When $n≥10$⁠, there are some open subsets $In,+$ of $s∊(0,n/2)$⁠, where $Zn(s;a1,…,an)$ is positive for all $(a1,…,an)∊(R+)n$⁠. By regarding $Zn(s;a1,…,an)$ as a function of $s$⁠, we find that when $n≥10$⁠, the generalized Riemann hypothesis is false for all $(a1,…,an)$⁠.