Multiplicative series, modular forms, and Mandelbrot polynomials

Larsen, Michael

doi:10.1090/mcom/3564

Multiplicative series, modular forms, and Mandelbrot polynomials
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by Michael Larsen; with an appendix by Anne Larsen HTML | PDF

Math. Comp. 90 (2021), 345-377 Request permission

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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