The Euler-Riemann Gases, and Partition Identities

The Euler theorem in partition theory and its generalization are derived from a non-interacting quantum field theory in which each bosonic mode with a given frequency is equivalent to a sum of bosonic mode whose frequency is twice ($s$-times) as much, and a fermionic (parafermionic) mode with the same frequency. Explicit formulas for the graded parafermionic partition functions are obtained, and the inverse of the graded partition function (IGPPF), turns out to be bosonic (fermionic) partition function depending on the parity of the order $s$ of the parafermions. It is also shown that these partition functions are generating functions of partitions of integers with restrictions. If the parity of the order $s$ is even, then mixing a system of parafermions with a system whose partition function is (IGPPF), results in a system of fermions and bosons. On the other hand, if the parity of $s$ is odd, then, the system we obtain is still a mixture of fermions and bosons but the corresponding Fock space of states is truncated. It turns out that these partition functions are given in terms of the Jacobi theta function $\theta_{4}$, and generate sequences in partition theory. Our partition functions coincide with the overpartitions, and jagged partitions in conformal field theory. Also, The partition functions obtained are related to the Ramond characters of the superconformal minimal models, and in the counting of the Moore-Read edge spectra that appear in the fractional quantum Hall effect. The different partition functions for the Riemann gas that are the counter parts of the Euler gas are obtained by a simple change of variables. In particular the counter part of the Jacobi Theta function is $\frac{\zeta(2t)}{\zeta(t)^2}$.