arXiv subjects

Jorge Zuniga

Publications and source records attributed to Jorge Zuniga.

3 recordsLinked to original sources

arXiv2026

Fast Ramanujan--type Series for Logarithms. Part II

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).

math.NT↗

arXiv2026

Fast Computing Formulas for some Dirichlet L-Series

For $χ_k$ a self$-$dual primitive Dirichlet character mod $k$ several reduced identities of Dirichlet $L-$functions $L_k(s):=L(s,χ_k)$, expressed as linear combinations of Hurwitz $ζ$ functions, are found for $s=2,3$ and some selected values of $k$. By using a merged approach between the Wilf$-$Zeilberger method and a Dougall$'$s $_5H_5$ technique, new proven accelerated series of hypergeometric$-$type are derived for specific Hurwitz $ζ$ function values. These fast series that are computed by means of the binary splitting algorithm, enter into the reduced identities found producing very efficient formulas to compute selected $L-$function values. The new algorithms include $L_k(2)$ for $k = -4$ Catalan's constant, $-7, -8, -15, -20, -24$ together with $L_k(3)$ for $k = 1$ Apery's constant, $5, 8$ and $12$. Formulas were tested and verified up to 100 million decimal places for each $L-$value.

math.NT↗

arXiv2025

Fast Ramanujan-type Series for Logarithms. Part I

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.

math.NT↗