  We show that the Green's function of a two dimensional fermion with a
modified dispersion relation and short distance parameter $a$ is given by the
Lerch zeta function. The Green's function is defined on a cylinder of radius R
and we show that the condition $R = a$ yields the Riemann zeta function as a
quantum transition amplitude for the fermion. We formulate the Riemann
hypothesis physically as a nonzero condition on the transition amplitude
between two special states associated with the point of origin and a point half
way around the cylinder each of which are fixed points of a $Z_2$
transformation. By studying partial sums we show that that the transition
amplitude formulation is analogous to neutrino mixing in a low dimensional
context. We also derive the thermal partition function of the fermionic theory
and the thermal divergence at temperature $1/a.$ In an alternative harmonic
oscillator formalism we discuss the relation to the fermionic description of
two dimensional string theory and matrix models. Finally we derive various
representations of the Green's function using energy momentum integrals, point
particle path integrals, and string propagators.
