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 analog of the Witten index, and use this, too, to obtain some number theoretic results related to the Riemann zeta function.