On the Minima and Convexity of Epstein Zeta Function

to the complex plane. We show that for fixed s 6= n/2, the function Zn(s; a1, . . . , an), as a function of (a1, . . . , an) ∈ (R) with fixed Qn i=1 ai, has a unique minimum at the point a1 = . . . = an. When Pn i=1 ci is fixed, the function (c1, . . . , cn) 7→ Zn (s; e c1 , . . . , en ) 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), 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). 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).