On the minima and convexity of Epstein Zeta function

S. C. Lim; L. P. Teo

doi:10.1063/1.2953513

paper

On the minima and convexity of Epstein Zeta function

arXiv:0806.1999 · doi:10.1063/1.2953513

Abstract

Let $Z_{n} (s; a_{1}, ..., a_{n})$ 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 $s \neq = n /2$ , 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 $1 \leq n \leq 9$ , $Z_{n} (s; a_{1}, ..., a_{n})$ as a function of $(a_{1}, ..., a_{n}) \in (R^{+})^{n}$ , can be both positive and negative for every $s \in (0, n /2)$ . When $n \geq 10$ , there are some open subsets $I_{n, +}$ of $s \in (0, n /2)$ , 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 $n \geq 10$ , the generalized Riemann hypothesis is false for all $(a_{1}, ..., a_{n})$ .

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On the minima and convexity of Epstein Zeta function

Abstract

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