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An efficient determination of the coefficients in the Chudnovskys’ series for 1/\(\pi \)

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Abstract

In 1914, Srinivasa Ramanujan published several hypergeometric series for \(1/\pi \). One of these series was used by Bill Gosper in 1985 in a world-record-computation of \(\pi \). Shortly after this, the Chudnovskys found a faster series for \(1/\pi \) based on the largest Heegner number and the Borweins proved Ramanujan’s series. Lately, the Chudnovskys’ series has often been used in practice to calculate digits of \(\pi \), it reads:

$$\begin{aligned} \frac{\sqrt{640320^3}}{12 \pi } = \sum _{n=0}^\infty \frac{\left( 6n\right) !}{\left( 3n\right) !\left( n!\right) ^3}\,\frac{13591409+ 545140134 n}{\left( -640320^3\right) ^n}. \end{aligned}$$

In this paper, we calculate the coefficients in two of Ramanujan’s series for \(1/\pi \) and those in the Chudnovskys’ series. For our calculation, we don’t require special software packages, but only the Fourier expansions of the Eisenstein series with a precision of \(\approx 20\) decimals. We also prove the exactness of our calculations by proving that the values of certain non-holomorphic modular functions are algebraic integers. Our proof uses the division values of the Weierstraß \(\wp \) function.

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Acknowledgements

For their invaluable help I am grateful to Henri Cohen, Michael Griffin, Jesús Guillera and the anonymous referee.

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Correspondence to Lorenz Milla.

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Milla, L. An efficient determination of the coefficients in the Chudnovskys’ series for 1/\(\pi \). Ramanujan J 57, 803–809 (2022). https://doi.org/10.1007/s11139-020-00330-6

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