  In this work we study the Plancherel-Rotach type asymptotics for
Stieltjes-Wigert orthogonal polynomials with complex scaling. The main term of
the asymptotics contains Ramanujan function $A_{q}(z)$ for the scaling
parameter on the vertical line $\Re(s)=2$, while the main term of the
asymptotics involves the theta functions for the scaling parameter in the strip
$0<\Re(s)<2$. In the latter case the number theoretical property of the scaling
parameter completely determines the order of the error term. $ $ These
asymptotic formulas may provide some insights to some new random matrix model
and also add a new link between special functions and number theory.
