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On the universality of the Epstein zeta ...
Andersson, Johan...

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On the universality of the Epstein zeta function by Andersson, Johan ( Author )
Lemonjar Open Access
N.A
24-08-2015
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 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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https://arxiv.org/abs/1508.05836
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