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 Sigma(n)(k is an element of Z)\{0}(Sigma(n)(i=1)[a(i)k(i)](2))(-s), Re s>n/2 to the complex plane. We show that for fixed s not equal n/2, the function Z(n)(s;a(1), ... ,a(n)) as a function of (a(1), ... ,a(n))is an element of(R+)(n) with fixed Pi(n)(i=1)a(i) has a unique minimum at the point a(1)= ... =a(n). When Sigma(n)(i=1)c(i) is fixed, the function (c(1), ... ,c(n)) bar right arrow Z(n)(s;e(1)(c), ... ,e(n)(c)) 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 1 <= n <= 9, Z(n)(s;a(1), ... ,a(n)) as a function of (a(1), ... ,a(n))is an element of(R+)(n) can be both positive and negative for every s is an element of(0,n/2). When n >= 10, there are some open subsets I-n,I-+ of s is an element of(0,n/2), where Z(n)(s;a(1), ... ,a(n)) is positive for all (a(1), ... ,a(n))is an element of(R+)(n). By regarding Z(n)(s;a(1), ... ,a(n)) as a function of s, we find that when n >= 10, the generalized Riemann hypothesis is false for all (a(1), ... ,a(n)). (C) 2008 American Institute of Physics.