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On the universality of the Epstein zeta function
Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, Utbildningsvetenskap och Matematik. (Mathematics and Applied Mathematics)
University of Copenhagen.
(engelsk)Manuskript (preprint) (Annet vitenskapelig)
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.

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matematik/tillämpad matematik
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URN: urn:nbn:se:mdh:diva-30034OAI: oai:DiVA.org:mdh-30034DiVA, id: diva2:885543
Tilgjengelig fra: 2015-12-18 Laget: 2015-12-18 Sist oppdatert: 2015-12-22bibliografisk kontrollert

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

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