Citation

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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Abstract

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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