Let Zn(s;a1,...,an) be the Epstein zeta function defined as the
meromorphic continuation of the function
\sum_{k\in\Z^n\setminus\{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=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).Comment: 27 page