Fast Ramanujan--type Series for Logarithms. Part II
Provides highly efficient formulas for computing logarithms of natural numbers, enabling record-breaking precision for log(10).
This paper presents novel hypergeometric formulas for computing logarithms, achieving the fastest known hypergeometric multiseries evaluation for selected integers. The method was used to compute log(10) to 2.0×10^12 decimal places.
This work extends the results of the preprint Ramanujan type Series for Logarithms, Part I, arXiv:2506.08245, which introduced single hypergeometric type identities for the efficient computing of $\log(p)$, where $p\in\mathbb{Z}_{>1}$. We present novel formulas for arctangents and methods for a very fast multiseries evaluation of logarithms. Building upon a $\mathcal{O}((p-1)^{6})$ Ramanujan type series asymptotic approximation for $\log(p)$ as $p\rightarrow1$, formulas for computing $n$ simultaneous logarithms are developed. These formulas are derived by solving an integer programming problem to identify optimal variable values within a finite lattice $\mathbb{Z}^{n}$. This approach yields linear combinations of series that provide: (i) highly efficient formulas for single logarithms of natural numbers (some of them were tested to get more than $10^{11}$ decimal places) and (ii) the fastest known hypergeometric formulas for multivalued logarithms of $n$ selected integers in $\mathbb{Z}_{>1}$. An application of these results was to extend the number of decimal places known for log(10) up to 2.0$\cdot$10$^{12}$ digits (June 06 2025).