Fast Ramanujan-type Series for Logarithms. Part I

TL;DR

This paper presents new hypergeometric series identities for rapidly computing logarithms of small integers, derived using WZ and LLL algorithms, enabling highly efficient and precise calculations suitable for software implementation.

Contribution

Introduction of novel hypergeometric series identities for logarithms, derived via WZ and LLL methods, with proven correctness and practical computational advantages.

Findings

Derived series achieve high convergence speed

Over 10^12 decimal places computed for some logs

Series are suitable for efficient software implementation

Abstract

This report introduces new series and variations of some hypergeometric type identities for fast computing of logarithms $\log\,p$ for small positive integers $p$. These series were found using Wilf Zeilberger (WZ) method and/or integer detection algorithms (LLL) providing highly efficient linearly convergent rational approximants for these constants. Some of the new identities are of $_4F_3$ type, but higher ones are found as well and hypergeometric series for log p, with variable p, have been derived. Found identities are proven by I. classical Beta Integral methods, II. some hypergometric closed forms and III. rational certificates from the WZ method. Since they are very fast, these series are particularly suitable to be embodied in mathematical software being implemented in binary splitting form which produces very efficient algorithms. Over 10e12 decimal places have been obtained for some logarithms in reasonable time.