  We find examples of duality among quantum theories that are related to
arithmetic functions by identifying distinct Hamiltonians that have identical
partition functions at suitably related coupling constants or temperatures. We
are led to this after first developing the notion of partial supersymmetry-in
which some, but not all, of the operators of a theory have superpartners-and
using it to construct fermionic and parafermionic thermal partition functions,
and to derive some number theoretic identities. In the process, we also find a
bosonic analogue of the Witten index, and use this, too, to obtain some number
theoretic results related to the Riemann zeta function.
