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:

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.

在1914年，Srinivasa Ramanujan发表了\（1 / \ pi \）的几个超几何级数。1985年，Bill Gosper在世界纪录\（\ pi \）中使用了其中一个系列。此后不久，Chudnovskys根据最大的Heegner数找到了一个更快的\（1 / \ pi \）级数，而Borweins证明了Ramanujan的级数。最近，在实践中经常使用Chudnovskys的系列来计算\（\ pi \）的位数，其内容为：

在本文中，我们计算了Ramanujan的两个\（1 / \ pi \）系数和Chudnovskys的系数。对于我们的计算，我们不需要特殊的软件包，而只需要Eisenstein系列的Fourier展开，精度为小数\（\大约20 \）。我们还通过证明某些非亚纯模函数的值是代数整数来证明我们计算的准确性。我们的证明使用了Weierstraß \（\ wp \）函数的除法值 。