Fast Ramanujan--type Series for Logarithms. Part II

TL;DR

This paper introduces new hypergeometric series formulas for efficiently computing logarithms and arctangents, achieving unprecedented precision and speed, including extending the known digits of log(10) to over 2 trillion.

Contribution

It develops novel Ramanujan-type series and an integer programming approach for fast, high-precision logarithm calculations, surpassing previous methods in efficiency and accuracy.

Findings

Achieved over 10^{11} decimal places for some logarithms.

Developed the fastest known hypergeometric formulas for multivalued logarithms.

Extended the known digits of log(10) to over 2 trillion.

Abstract

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).