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On the universality...
On the universality of the Epstein zeta function
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- Andersson, Johan (author)
- Örebro universitet,Örebro University
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- Södergren, Anders, 1979 (author)
- Chalmers tekniska högskola,Chalmers University of Technology,Göteborgs universitet,University of Gothenburg
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(creator_code:org_t)
- 2020
- 2020
- English.
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In: Commentarii Mathematici Helvetici. - 0010-2571 .- 1420-8946. ; 95:1, s. 183-209
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Abstract
Subject headings
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- We study universality properties of the Epstein zeta function E-n(L,s) for lattices L of large dimension n and suitable regions of complex numbers s. Our main result is that, as n -> infinity, E-n(L,s) is universal in the right half of the critical strip as L varies over all n-dimensional lattices L. The proof uses a novel combination of an approximation result for Dirichlet polynomials, 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 approach we also prove that, as n -> infinity, E-n(L-1,s) - E-n(L-2,s) is universal in the full half-plane to the right of the critical line as E-n(L,s) varies over all pairs of n-dimensional lattices. Finally, we prove a more classical universality result for E-n(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 E-n(L,s) on the critical line that is subconvex for n >= 5 and almost all n-dimensional lattices L.
Subject headings
- NATURVETENSKAP -- Matematik -- Beräkningsmatematik (hsv//swe)
- NATURAL SCIENCES -- Mathematics -- Computational Mathematics (hsv//eng)
- NATURVETENSKAP -- Matematik -- Diskret matematik (hsv//swe)
- NATURAL SCIENCES -- Mathematics -- Discrete Mathematics (hsv//eng)
- NATURVETENSKAP -- Matematik -- Matematisk analys (hsv//swe)
- NATURAL SCIENCES -- Mathematics -- Mathematical Analysis (hsv//eng)
Keyword
- subconvexity
- universality
- Poisson process
- random lattice
- Epstein zeta function
Publication and Content Type
- art (subject category)
- ref (subject category)
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