## Multiplicative series, modular forms, and Mandelbrot polynomials

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Michael Larsen; with an appendix by Anne Larsen
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## Abstract:

We say a power series $\sum _{n=0}^\infty a_n q^n$ is*multiplicative*if the sequence $1,a_2/a_1,\ldots ,a_n/a_1,\ldots$ is so. In this paper, we consider multiplicative power series $f$ such that $f^2$ is also multiplicative. We find a number of examples 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 $\mathbb {C}$. The precise determination of this variety turns out to be a finite computational problem, but it seems to be beyond 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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## Additional Information

**Michael Larsen**- Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
- MR Author ID: 293592
- Email: mjlarsen@indiana.edu
- Received by editor(s): October 29, 2019
- Received by editor(s) in revised form: April 4, 2020
- Published electronically: September 9, 2020
- Additional Notes: The author was partially supported by the Sloan Foundation and the NSF
- © Copyright 2020 American Mathematical Society
- Journal: Math. Comp.
**90**(2021), 345-377 - MSC (2020): Primary 11F25; Secondary 37F46, 30C85, 11F27
- DOI: https://doi.org/10.1090/mcom/3564
- MathSciNet review: 4166464