Lattice sums of hyperplane arrangements

TL;DR

This paper introduces lattice sums related to hyperplane arrangements, explores their properties using convex polytope theory, and evaluates special values including zeta-functions of root systems and their affine analogues.

Contribution

It generalizes zeta-functions of root systems through lattice sums associated with hyperplane arrangements and develops methods to evaluate their special values.

Findings

Derived explicit evaluations of lattice sums and zeta-functions.

Connected lattice sums to convex polytope theory.

Provided new insights into hyperplane arrangement zeta-functions.

Abstract

We introduce certain lattice sums associated with hyperplane arrangements, which are (multiple) sums running over integers, and can be regarded as generalizations of certain linear combinations of zeta-functions of root systems. We also introduce generating functions of special values of those lattice sums, and study their properties by virtue of the theory of convex polytopes. Consequently we evaluate special values of those lattice sums, especially certain special values of zeta-functions of root systems and their affine analogues. In some special cases it is possible to treat sums running over positive integers, which may be regarded as zeta-functions associated with hyperplane arrangements.