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On the universality of the Epstein zeta function
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. (Mathematics and Applied Mathematics)
University of Copenhagen.
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We study universality properties of the Epstein zeta function En(L,s) for lattices L of large dimension n and suitable regions of complex numbers s . Our main result is that, as n→∞ , En(L,s) is universal in the right half of the critical strip as L varies over all n -dimensional lattices L . The proof uses an approximation result for Dirichlet polynomials together with a recent result on the distribution of lengths of lattice vectors in a random lattice of large dimension and a strong uniform estimate for the error term in the generalized circle problem. Using the same basic approach we also prove that, as n→∞ , En(L1,s)−En(L2,s) is universal in the full half-plane to the right of the critical line as (L1,L2) varies over all pairs of n -dimensional lattices. Finally, we prove a more classical universality result for En(L,s) in the s -variable valid for almost all lattices L of dimension n . As part of the proof we obtain a strong bound of En(L,s) on the critical line that is subconvex for n≥5 and almost all n -dimensional lattices L.

National Category
Mathematical Analysis
Research subject
Mathematics/Applied Mathematics
Identifiers
URN: urn:nbn:se:mdh:diva-30034OAI: oai:DiVA.org:mdh-30034DiVA: diva2:885543
Available from: 2015-12-18 Created: 2015-12-18 Last updated: 2015-12-22Bibliographically approved

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http://arxiv.org/pdf/1508.05836v1.pdf

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