  We show that the chemical potential of a one-dimensional (1D) interacting
Bose gas exhibits a non-monotonic temperature dependence which is peculiar of
superfluids. The effect is a direct consequence of the phononic nature of the
excitation spectrum at large wavelengths exhibited by 1D Bose gases. For low
temperatures $T$, we demonstrate that the coefficient in $T^2$ expansion of the
chemical potential is entirely defined by the zero-temperature density
dependence of the sound velocity. We calculate that coefficient along the
crossover between the Bogoliubov weakly-interacting gas and the Tonks-Girardeau
gas of impenetrable bosons. Analytic expansions are provided in the asymptotic
regimes. The theoretical predictions along the crossover are confirmed by
comparison with the exactly solvable Yang-Yang model in which the
finite-temperature equation of state is obtained numerically by solving
Bethe-{\it ansatz} equations. A 1D ring geometry is equivalent to imposing
periodic boundary conditions and arising finite-size effects are studied in
details. At $T=0$ we calculated various thermodynamic functions, including the
inelastic structure factor, as a function of the number of atoms, pointing out
the occurrence of important deviations from the thermodynamic limit.
