  We introduce the Hadamard topology on the Witt ring of rational functions,
giving a simultaneous refinement of the weight and point-counting topologies.
Zeta functions of algebraic varieties over finite fields are elements of the
rational Witt ring, and the Hadamard topology allows for a conjectural
unification of results in arithmetic and motivic statistics: The completion of
the Witt ring for the Hadamard topology can be identified with a space of
meromorphic functions which we call Hadamard functions, and we make the
meta-conjecture that any "natural" sequence of zeta functions which converges
to a Hadamard function in both the weight and point-counting topologies
converges also in the Hadamard topology. For statistics arising from Bertini
problems, zero-cycles or the Batyrev-Manin conjecture, this yields an explicit
conjectural unification of existing results in motivic and arithmetic
statistics that were previously connected only by analogy. As evidence for our
conjectures, we show that Hadamard convergence holds for many natural
statistics arising from zero-cycles, as well as for the motivic height zeta
function associated to the motivic Batyrev-Manin problem for split toric
varieties.

Algebraic Geometry;Number Theory