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 fun...
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AMER INST PHYSICS
2008
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| Online Access: | http://shdl.mmu.edu.my/2299/ |
| _version_ | 1848790018966945792 |
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| author | Lim, S. C. Teo, L. P. |
| author_facet | Lim, S. C. Teo, L. P. |
| author_sort | Lim, S. C. |
| building | MMU Institutional Repository |
| collection | Online Access |
| description | 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. |
| first_indexed | 2025-11-14T18:05:57Z |
| format | Article |
| id | mmu-2299 |
| institution | Multimedia University |
| institution_category | Local University |
| last_indexed | 2025-11-14T18:05:57Z |
| publishDate | 2008 |
| publisher | AMER INST PHYSICS |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | mmu-22992011-08-24T06:15:13Z http://shdl.mmu.edu.my/2299/ On the minima and convexity of Epstein zeta function Lim, S. C. Teo, L. P. T Technology (General) QC Physics 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. AMER INST PHYSICS 2008-07 Article NonPeerReviewed 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 http://dx.doi.org/10.1063/1.2953513 doi:10.1063/1.2953513 doi:10.1063/1.2953513 |
| spellingShingle | T Technology (General) QC Physics Lim, S. C. Teo, L. P. On the minima and convexity of Epstein zeta function |
| title | On the minima and convexity of Epstein zeta function |
| title_full | On the minima and convexity of Epstein zeta function |
| title_fullStr | On the minima and convexity of Epstein zeta function |
| title_full_unstemmed | On the minima and convexity of Epstein zeta function |
| title_short | On the minima and convexity of Epstein zeta function |
| title_sort | on the minima and convexity of epstein zeta function |
| topic | T Technology (General) QC Physics |
| url | http://shdl.mmu.edu.my/2299/ http://shdl.mmu.edu.my/2299/ http://shdl.mmu.edu.my/2299/ |