  We say a power series $\sum_{n=0}^\infty a_n q^n$ is multiplicative if the
function $n\mapsto a_n/a_1$ ($n\ge 1$) is so. In this paper, we consider
multiplicative power series $f$ such that $f^2$ is also multiplicative. We find
various solutions for which $f$ is a rational function or a theta series and
prove that the complete set of solutions is the locus of a (probably reducible)
affine variety over C. The precise determination of this variety is a finite
computational problem but seems to be outside the reach of current computer
algebra systems. The proof of the theorem depends on a bound on the logarithmic
capacity of the Mandelbrot set.

Number Theory