On the universality of the Epstein zeta function

Abstract

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.Let us also note that, since ζ(s) has an Euler product, it is necessary to assume that the function f in Theorem 1.1 is nonvanishing on K. However, for zeta functions

Epstein Zeta-Functions, Subconvexity, and the Purity Conjecture

“…Even in a compact part of the manifold, however, large sup-norms can occur when φ is a lift coming from a smaller group. This was first observed by Rudnick and Sarnak [23] for theta-lifts on arithmetic 3-manifolds, based on similar phenomena of Eisenstein series on SO (3,1). This has been extended and generalized in [9,21]; in particular, for each n 5 there exist compact n-dimensional hyperbolic manifolds of constant negative curvature having a sequence of…”

“…(2) Corollary 4 exhibits explicitly a non-zero accumulation point, which in the case of odd k fails to be in 1 4 Z. Therefore, it appears that a proper generalization of Sarnak's purity conjecture for the continuous spectrum may depend on the degree of degeneracy one allows for the automorphic forms under consideration, and in particular on the parabolic subgroup with which an Eisenstein series is associated.…”

“…We start from the approximate functional equation (2.2). Let W be a fixed smooth function with support in [1,2]. For the proof of (1.5) it suffices (by the Cauchy-Schwarz inequality) to show that…”

Voronin Universality in several complex variables

“…When both α n and α n−1 are rational we get a smaller region where we have universality 6 unless we assume the Riemann hypothesis for some Dirichlet L-functions. For example for n = 2 and α 1 = α 2 = 1 which is the Euler-Zagier case we are sure to get universality in the region where 3/4 < Re(s j ) < 1 rather than 1/2 < Re(s j ) < 1 for j = 1, 2.…”

A spectral universality theorem for Maass L-functions

“…Note that universality results for L-functions analogous to (1.1) do not hold on the left of the critical line σ = ℜ(s) = 1/2, due to the restrictions imposed by the functional equation, nor on the right of the line of absolute convergence σ = 1, due to boundedness properties; see the discussion in Perelli-Righetti [16] for further information. Interestingly enough, there exists a different type of universality that hold also on the right of σ = 1, see Andersson-Södergren [1,Theorem 1.10].…”