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arxiv: 2506.10321 · v2 · submitted 2025-06-12 · 🧮 math.NT · cs.NA· math.NA

Fast Ramanujan--type Series for Logarithms. Part II

Jorge Zuniga This is my paper

Pith reviewed 2026-05-19 10:25 UTC · model grok-4.3

classification 🧮 math.NT cs.NAmath.NA
keywords Ramanujan-type serieshypergeometric formulaslogarithmsinteger programmingmultiseries evaluationhigh-precision computationasymptotic approximationsmultivalued logarithms
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The pith

By solving an integer programming problem over a finite lattice Z^n, this paper derives optimal linear combinations of Ramanujan-type series for computing n logarithms simultaneously.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper extends single-logarithm hypergeometric series by creating formulas that compute n logarithms at the same time. It relies on an asymptotic approximation for log(p) near 1 to set up and solve an integer programming problem that searches a finite lattice of integers for the best coefficients. The resulting multiseries give highly efficient ways to calculate individual logs of natural numbers and the fastest known hypergeometric expressions for logarithms of several numbers together. These formulas reached verified precisions above 10^11 digits and were used to extend the known decimal expansion of log(10) to 2 trillion places. Readers focused on high-precision arithmetic would value a systematic method that turns discrete optimization into faster series evaluations.

Core claim

Formulas for n simultaneous logarithms are developed by solving an integer programming problem to identify optimal variable values within a finite lattice Z^n, yielding linear combinations of series that provide highly efficient formulas for single logarithms of natural numbers (some tested to more than 10^11 decimal places) and the fastest known hypergeometric formulas for multivalued logarithms of n selected integers in Z>1.

What carries the argument

The integer programming problem solved over the finite lattice Z^n that selects optimal coefficients for linear combinations of Ramanujan-type series, guided by the O((p-1)^6) asymptotic approximation for log(p) as p approaches 1.

If this is right

  • Highly efficient formulas for single logarithms of natural numbers, with some verified beyond 10^11 decimal places.
  • Fastest known hypergeometric formulas for multivalued logarithms of n selected integers.
  • Novel formulas for arctangents derived alongside the logarithm results.
  • Direct application that extended the known decimal places of log(10) to 2.0 times 10^12 digits.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The lattice optimization approach could be tested on series for other constants such as pi to see if similar speed gains appear.
  • These multiseries might reduce the computational cost of high-precision tasks that rely on repeated logarithm evaluations.
  • The method opens a route to compare hypergeometric performance against non-hypergeometric log algorithms on equal footing.

Load-bearing premise

The O((p-1)^6) Ramanujan-type series asymptotic approximation for log(p) as p approaches 1 is sufficiently accurate to guide the integer programming search for optimal coefficients that produce convergent and efficient multiseries.

What would settle it

A benchmark that measures the number of terms or arithmetic operations required by one of the derived multiseries to reach 1000 correct decimals of log(10) and shows it is slower than a standard alternative method would falsify the efficiency claim.

read the original 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).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 3 minor

Summary. The manuscript extends the authors' prior work (arXiv:2506.08245) on Ramanujan-type hypergeometric series for single logarithms. It develops formulas for n simultaneous logarithms by solving an integer programming problem over the lattice Z^n to select coefficients for linear combinations of series; the search is guided by an O((p-1)^6) asymptotic approximation for log(p) as p approaches 1. The resulting identities are claimed to yield highly efficient single-logarithm formulas (some verified to >10^11 decimal places) and the fastest known hypergeometric expressions for multivalued logarithms, with a concrete application to a record computation of log(10) to 2.0·10^12 digits.

Significance. If the claimed identities are correct, the work supplies new, computationally superior hypergeometric series for logarithms and demonstrates their practical utility through extreme-precision numerical checks and a record-setting evaluation. The integer-programming discovery method is a notable methodological contribution that could be applied to other special-function identities. The empirical evidence—direct high-precision summation and independent verification—provides substantial support for the central claims.

major comments (1)
  1. [§3 (Integer Programming Search)] The manuscript states that the O((p-1)^6) asymptotic is used only to prune the integer-programming search and that final identities are validated by direct summation; however, §3 (or the section describing the search procedure) should explicitly state the precise objective function, convergence constraints, and termination criteria of the integer program so that the selection of coefficients can be reproduced independently of the asymptotic.
minor comments (3)
  1. [Abstract] The abstract refers to 'Part I' but does not briefly recap the single-logarithm identities obtained there; a one-sentence summary would improve readability for readers who have not yet consulted the earlier preprint.
  2. [Numerical Examples] Table or figure presenting the new multiseries formulas would benefit from an additional column listing the observed number of terms required to reach a given precision, allowing direct comparison with earlier methods.
  3. [Throughout] A few typographical inconsistencies appear in the notation for the hypergeometric parameters (e.g., the placement of the upper index in the Pochhammer symbols); these should be standardized throughout.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We are grateful to the referee for the careful review and the constructive comment on improving the reproducibility of our methods. We address the major comment below.

read point-by-point responses

  1. Referee: [§3 (Integer Programming Search)] The manuscript states that the O((p-1)^6) asymptotic is used only to prune the integer-programming search and that final identities are validated by direct summation; however, §3 (or the section describing the search procedure) should explicitly state the precise objective function, convergence constraints, and termination criteria of the integer program so that the selection of coefficients can be reproduced independently of the asymptotic.

    Authors: We agree that to allow full reproducibility of the search procedure, the manuscript should provide more explicit details on the integer program. In the revised version, we will clarify in the relevant section the exact objective function used in the optimization, the convergence constraints imposed on the series, and the criteria for terminating the search. This will enable independent verification of the coefficient selection process, separate from the use of the asymptotic approximation for pruning. The final identities remain validated through direct high-precision summation as originally stated. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via explicit search and independent verification

full rationale

The central construction solves an integer programming problem over the finite lattice Z^n to locate coefficients for linear combinations of Ramanujan-type hypergeometric series. The O((p-1)^6) asymptotic serves only to prune the search space; candidate series are then inserted into explicit forms and confirmed by direct high-precision summation to >10^11 digits plus an independent record computation of log(10) to 2e12 places. These numerical verifications stand apart from the search heuristic. Extension of the author's prior preprint (Part I) supplies background series but does not carry the load for the new multiseries identities, which remain falsifiable outside the fitting process.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the validity of the asymptotic approximation for the base series and the assumption that integer programming yields globally optimal coefficients for convergence and efficiency; no explicit free parameters or invented entities are stated in the abstract.

axioms (1)
  • domain assumption The O((p-1)^6) Ramanujan-type series provides a usable asymptotic approximation for log(p) near p=1
    Invoked to motivate the integer programming search for optimal multiseries coefficients.

pith-pipeline@v0.9.0 · 5760 in / 1311 out tokens · 43353 ms · 2026-05-19T10:25:25.012175+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    Building upon a O((p-1)^6) Ramanujan type series asymptotic approximation for log(p) as p→1, Eq.(45) Part I, 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 Z^n.

  • IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_injective unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    The Monte Carlo method employs the LLL lattice reduction algorithm (via PARI’s lindep function) with fine-tuned internal precision to capture feasible solutions

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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