On covering by translates of a set

In this paper we study the minimal number of translates of an arbitrary subset $S$ of a group $G$ needed to cover the group, and related notions of the efficiency of such coverings. We focus mainly on finite subsets in discrete groups, reviewing the classical results in this area, and generalizing them to a much broader context. For example, we show that while the worst-case efficiency when $S$ has $k$ elements is of order $1/\log k$, for $k$ fixed and $n$ large, almost every $k$-subset of any given $n$-element group covers $G$ with close to optimal efficiency.Comment: 41 pages; minor corrections; to appear in Random Structures and Algorithm

The Integral of the Riemann ξ-Function

ABSTRACT. This paper studies the integral of the Riemann ξ-function defined by ξ(−1)(s) = ∫ s 1/2 ξ(w)dw. More generally, it studies a one-parameter family of func-tions given by Fourier integrals and satisfying a functional equation. Members of this family are shown to have only finitely many zeros on the critical line, with ξ(−1)(s) having exactly one zero on the critical line, at s = 12. It is also shown there are ze-ros of ξ(−1)(s) that lie arbitrarily far away from the critical line. An analogue of the de-Bruijn-Newman constant is introduced for this family, and shown to be infinite. 1