  We apply noise thermometry to characterize charge and thermoelectric
transport in single InAs nanowires (NWs) at a bath temperature of 4.2 K. Shot
noise measurements identify elastic diffusive transport in our NWs with
negligible electron-phonon interaction. This enables us to set up a measurement
of the diffusion thermopower. Unlike in previous approaches, we make use of a
primary electronic noise thermometry to calibrate a thermal bias across the NW.
In particular, this enables us to apply a contact heating scheme, which is much
more efficient in creating the thermal bias as compared to conventional
substrate heating. The measured thermoelectric Seebeck coefficient exhibits
strong mesoscopic fluctuations in dependence on the back-gate voltage that is
used to tune the NW carrier density. We analyze the transport and
thermoelectric data in terms of approximate Mott's thermopower relation and to
evaluate a gate-voltage to Fermi energy conversion factor.
