Multiplicative Series, Modular Forms, and Mandelbrot Polynomials

TL;DR

This paper investigates special multiplicative power series whose squares are also multiplicative, characterizing their solutions as points on an affine variety and linking the problem to complex dynamics via the Mandelbrot set.

Contribution

It classifies multiplicative power series with multiplicative squares, identifying solutions as rational functions or theta series and describing their solution space as an affine variety.

Findings

Solutions include rational functions and theta series.

The solution set forms an affine variety over C.

The proof involves bounds related to the Mandelbrot set.

Abstract

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.