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arXiv · 1908.09974

Multiplicative Series, Modular Forms, and Mandelbrot Polynomials

Michael Larsen
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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.

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Michael Larsen. 2019-10-29. Multiplicative Series, Modular Forms, and Mandelbrot Polynomials. https://arxiv.org/abs/1908.09974

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