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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Official URL: http://dx.doi.org/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 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,+) 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.

Item Type: Article
Subjects: T Technology > T Technology (General)
Q Science > QC Physics
Divisions: Faculty of Engineering and Technology (FET)
Depositing User: Ms Suzilawati Abu Samah
Date Deposited: 24 Aug 2011 06:15
Last Modified: 24 Aug 2011 06:15
URI: http://shdl.mmu.edu.my/id/eprint/2299

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