  Character expansion developed in real space renormalization group (RSRG)
approach is applied to U(1) lattice gauge theory with $\th$-term in 2
dimensions. Topological charge distribution $P(Q)$ is shown to be of Gaussian
form at any $\b$(inverse coupling constant). The partition function $Z(\th)$ at
large volume is shown to be given by the elliptic theta function. It provides
the information of the zeros of partition function as an analytic function of
$\ze= e^{i \th}$ ($\th$ = theta parameter). These partition function zeros lead
to the phase transition at $\th=\pi$. Analytical results will be compared with
the MC simulation results. In MC simulation, we adopt (i)``set method" and
(ii)``trial function method".
