Andreas Strömbergsson

Rogers' mean value formula: some applications and extensions

In 1955 C. A. Rogers proved an explicit formula for the expected value of a k-fold sum over a lattice L against an arbitrary test function, as L is taken random in the space of d-dimensional unimodular lattices, equipped with its invariant probability measure. This formula has found a number of applications over the years and I will describe some of these. In particular I will discuss the error term in the generalized circle problem for a random lattice in large dimension, and consequences regarding the value distribution of the Epstein zeta function along the real line. I will also describe an extension of Rogers' formula to a situation involving both L and its dual lattice L^*. The talk is based on joint work with Anders Södergren.