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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


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


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:
  • 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:
  • MathSciNet review: 4166464