Computing isolated coefficients of the $j$-function

Fredrik Johansson

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arXiv · 2011.14671

Computing isolated coefficients of the $j$-function

Fredrik Johansson

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Abstract

We consider the problem of efficiently computing isolated coefficients $c_n$ in the Fourier series of the elliptic modular function $j(\tau)$. We show that a hybrid numerical-modular method with complexity $n^{1+o(1)}$ is efficient in practice. As an application, we locate the first few values of $c_n$ that are prime, the first occurring at $n = 457871$.

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Fredrik Johansson. 2020-11-30. Computing isolated coefficients of the $j$-function. https://arxiv.org/abs/2011.14671

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