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.

中文翻译：

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乘法级数、模形式和 Mandelbrot 多项式

如果函数 $n\mapsto a_n/a_1$ ($n\ge 1$) 是乘法，我们说幂级数 $\sum_{n=0}^\infty a_n q^n$ 是乘法的。在本文中，我们考虑乘幂级数 $f$，使得 $f^2$ 也是乘法的。我们找到了各种解，其中 $f$ 是有理函数或 theta 级数，并证明完整的解集是 C 上的（可能可约化的）仿射变体的轨迹。这个变体的精确确定是一个有限的计算问题但似乎超出了当前计算机代数系统的范围。该定理的证明取决于 Mandelbrot 集的对数容量的界限。