ON THE EPSTEIN ZETA FUNCTION

On the Universality of the Epstein Zeta Function

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 d...

On the Minima and Convexity of Epstein Zeta Function

to the complex plane. We show that for fixed s 6= n/2, the function Zn(s; a1, . . . , an), as a function of (a1, . . . , an) ∈ (R) with fixed Qn i=1 ai, has a unique minimum at the point a1 = . . . = an. When Pn i=1 ci is fixed, the function (c1, . . . , cn) 7→ Zn (s; e c1 , . . . , en ) can be shown to be a convex function of any (n−1) of the variables {c1, . . . , cn}. These results are then ...

Sphere Packing, Lattices, and Epstein Zeta Function

On the Value Distribution of the Epstein Zeta Function in the Critical Strip

We study the value distribution of the Epstein zeta function En(L, s) for 0 < s < n 2 and a random lattice L of large dimension n. For any fixed c ∈ ( 1 4 , 1 2 ) and n → ∞, we prove that the random variable V −2c n En(·, cn) has a limit distribution, which we give explicitly (here Vn is the volume of the ndimensional unit ball). More generally, for any fixed ε > 0 we determine the limit distri...

On the Location of the Zero-free Half-plane of a Random Epstein Zeta Function

In this note we study, for a random lattice L of large dimension n, the supremum of the real parts of the zeros of the Epstein zeta function En(L, s) and prove that this random variable has a limit distribution, which we give explicitly. This limit distribution is studied in some detail; in particular we give an explicit formula for its distribution function.