  The partition function of a massless scalar field on a Euclidean spacetime
manifold $\mathbb{R}^{d-1}\times\mathbb{T}^2$ and with momentum operator in the
compact spatial dimension coupled through a purely imaginary chemical potential
is computed. It is modular covariant and admits a simple expression in terms of
a real analytic SL$(2,\mathbb{Z})$ Eisenstein series with $s=(d+1)/2$.
Different techniques for computing the partition function illustrate
complementary aspects of the Eisenstein series: the functional approach gives
its series representation, the operator approach yields its Fourier series,
while the proper time/heat kernel/world-line approach shows that it is the
Mellin transform of a Riemann theta function. High/low temperature duality is
generalized to the case of a non-vanishing chemical potential. By clarifying
the dependence of the partition function on the geometry of the torus, we
discuss how modular covariance is a consequence of full SL$(2,\mathbb{Z})$
invariance. When the spacetime manifold is
$\mathbb{R}^p\times\mathbb{T}^{q+1}$, the partition function is given in terms
of a SL$(q+1,\mathbb{Z})$ Eisenstein series again with $s=(d+1)/2$. In this
case, we obtain the high/low temperature duality through a suitably adapted
dual parametrization of the lattice defining the torus. On $\mathbb{T}^{d+1}$,
the computation is more subtle. An additional divergence leads to an harmonic
anomaly.
