On the minima and convexity of Epstein zeta function


Lim, S. C. and Teo, L. P. (2008) On the minima and convexity of Epstein zeta function. Journal of Mathematical Physics, 49 (7). 073513. ISSN 00222488

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Let Zn(s;a1,…,an) be the Epstein zeta function defined as the meromorphic continuation of the function math(∑i = 1n[aiki]2)−s,  Re s>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) ∊ (math+)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) ∊ (math+)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) ∊ (math+)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).

Item Type: Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Computing and Informatics (FCI)
Depositing User: Ms Suzilawati Abu Samah
Date Deposited: 04 Sep 2013 05:57
Last Modified: 04 Sep 2013 05:57
URII: http://shdl.mmu.edu.my/id/eprint/3928


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