Fast Ramanujan-type Series for Logarithms. Part I

2025/06/09 by Zuniga, Jorge
#11Y16 11Y60 33C20 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2506.08245

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 ₄F₃ 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.

Citations

• The WZ method and flawless WZ pairs
• Computing elementary functions using multi-prime argument reduction
• A simple algorithm for the evaluation of the hypergeometric series using quasi-linear time and linear space
• Hypergeometric Series Acceleration Via the WZ method
• Factoring polynomials with rational coefficients
• Analysis of PSLQ, an integer relation finding algorithm

Related

• Fast Ramanujan--type Series for Logarithms. Part II
• Logarithmic A-hypergeometric series
• Logarithmic Integrals: A Review from Gradshteyn and Ryzhik to Recent\n Times
• Ramanujan's formula for the logarithmic derivative of the gamma function
• Length of the Continued Logarithm Algorithm on Rational Inputs