On the minima and convexity of Epstein Zeta function

Let $Z_n(s; a_1,..., a_n)$ be the Epstein zeta function defined as the meromorphic continuation of the function sum_{kinZ^nsetminus{0}}(sum_{i=1}^n [a_i k_i]^2)^{-s}, text{Re} s>frac{n}{2} to the complex plane. We show that for fixed $s eq n/2$, the function $Z_n(s; a_1,..., a_n)$, as a function of $(a_1,..., a_n)in (R^+)^n$ with fixed $prod_{i=1}^n a_i$, has a unique minimum at the point $a_1=...=a_n$. When $sum_{i=1}^n c_i$ is fixed, the function $$(c_1,..., c_n)mapsto Z_n(s; e^{c_1},..., e^{c_n})$$ can be shown to be a convex function of any $(n-1)$ of the variables ${c_1,...,c_n}$. These results are then applied to the study of the sign of $Z_n(s; a_1,..., a_n)$ when $s$ is in the critical range $(0, n/2)$. It is shown that when $1leq nleq 9$, $Z_n(s; a_1,..., a_n)$ as a function of $(a_1,..., a_n)in (R^+)^n$, can be both positive and negative for every $sin (0,n/2)$. When $ngeq 10$, there are some open subsets $I_{n,+}$ of $sin(0,n/2)$, where $Z_{n}(s; a_1,..., a_n)$ is positive for all $(a_1,..., a_n)in(R^+)^n$. By regarding $Z_n(s; a_1,..., a_n)$ as a function of $s$, we find that when $ngeq 10$, the generalized Riemann hypothesis is false for all $(a_1,...,a_n)$.