irwin

[[TOC]]

Description

This repository is to document implementations of formulas from my pubication Measures for the summation of Irwin series, Integers 26:A11 (2026).

The above paper explains a novel approach to a venerable topic which got started in 1914 with A Curious Convergent Series by A. J. Kempner, and then was continued in 1916 with an article with the same title by F. Irwin.

Let $b>1$ be an integer (called base or radix), $d\in{0,...,b-1}$ a digit, $k$ a non-negative integer. We want to compute $S(b,d,k)$ which is the (convergent) sub-series of the harmonic series $\sum\frac{1}{n}$ where only those positive integers $n$ are kept whose radix-$b$ representation contains exactly $k$ occurrences of the digit $d$. Kempner and Irwin considered only $b=10$ (decimal system). Irwin added the parameter $k$ where Kempner had considered only excluding a given digit (i.e. $k=0$). Further generalizations are found in the litterature, see the Bibliographical references section.

To compute such quantities with the help of the SageMath code provided in this repository, start a SageMath session, enter load("irwin.sage") and then use either irwin() or irwinpos() procedure. For example:

sage: load("irwin.sage")
[some info printed]
sage: irwin(10,9,0,52)
22.92067661926415034816365709437593191494476243699848

The first argument is the radix $b$, the second the special digit $d$, the third is the number of occurrences $k$, and the fourth is the number of significant figures asked for.

Hundreds or thousands of decimal digits are computed easily:

sage: load("irwin.sage")
[some info printed mentioning maxworkers which defaults to 8]
sage: irwin(10,9,0,1002)
[the output is reformatted here to show 50 decimals per line]
22.92067661926415034816365709437593191494476243699848
   15685419983565721563381899111294456260374482018989
   90964125332346922160471190478310297506146968857121
   01806786493339402886962779578685961198637905620169
   32188040880170136179021106286611735099211021080576
   70378581471208344258765832272657620103831470760370
   30815999623544735896526905676888497081960327431233
   14588927997290413878495249814944204592152773507367
   07218520004083026308916169121123862636859589823575
   17170592498667879488473210892480659162340101523560
   00506548043749678309013031335561096953014813317749
   55762523805629716085009843545476018253422157510734
   48392165782984461954239160106117835383539414385364
   56085452218993239443664387904158857609144227813991
   99992242055353569500690341681751890944480911928277
   83446999651712608600666360667788028808406885936480
   28751790909188136795127797348003365941380076337136
   20275923523021897838806069615932191066192832138116
   95786715012908593756769518010810881852946961772722
   23692633510303284693132263332046629826719621921950
sage: irwin(10,9,1,1002)
[the output is reformatted here to show 50 decimals per line]
23.04428708074784831967594930973617482538959203064773
   62135578783008262042579280261007145671482118830782
   57921943364250671219896744152423420975036205340023
   04185784253363785559895004750435973037013624248692
   81693758690484874474533229028198473110373901313573
   11432258255340503360367871022252892752126397786453
   83974571210767152137049460532051975462039681885418
   36330606303719288329589891730925547573307176245910
   08049079306964817292360040758680171370594059804295
   02478333846389429682664501235076520170203860074879
   38353118391668568399204076846814722810630157329244
   15926574398873339404075981049839733114335471459356
   69038199984752130884656309345224669665996152072319
   08005559300473069297421785342383181330939055266000
   60305950156550830604733577855412791540396401495913
   50606434552699589093508098625808576790394619236238
   07758802238788028673496735651848096113580106792051
   15685105510116478927387735732377302778507573807355
   85556733655125237297349752705490397573629470971925
   89207555878085304702890838585263137514388675244390

Check help(irwin) for additional parameters.

Additional details

This repository is mainly there in order to make the software available. It does provide some additional mathematical content (see files irwin_v5_doc.pdf and taille_pascal.pdf), which is in relation with implementation choices or constraints, but for an introduction to the actual underlying core formulas, start with burnolmath.gitlab.io/irwin.

• irwin.sage is a symlink to irwin_v5.sage which is the latest installment of my code, as described further below. This is the file you should use preferentially to compute thousands of decimal digits.

  [!note] Notice also that irwin_v5.sage has internal code comments covering all key parts of the implementation, which is not the case of irwin_v3.sage.

• irwinfloat_legacy.py and irwin_legacy.sage are the same files as initially joined (with different names) to my arXiv paper v1 and last updated on April 6, 2024.

• irwin_v3.sage is a 2025 evolution of irwin_legacy.sage which is better suited to computing thousands of digits (say, starting at around 2000 digits).

  • it considerably reduces the memory footprint (for mathematical background related to the storage size of the binomial coefficients, see taille_pascal.pdf),
  • and has gains from using multiple precisions for the RealField objects (i.e. it computes tail terms with lower precision).
  • it uses by default level=3 which is better for getting thousands of digits.

  Compared to the 2024 legacy implementation, the manner of estimating how many terms of the Burnol series should be used has been made a bit safer (i.e. also for $k>0$ or if using level=4). Note though: as the code does not check in dropping guard digits if the value is very near a half-way point, to be absolutely certain to get the correctly rounded value at a given precision, one should do the computation asking for some extra digits (say 5 more)...

• irwin_v5.sage uses "parallelization". This is easy to do and efficient for the computation of the beta's.

  [!note] The quantities beta's (which depend on a basis $b$, a number of digits $l$, a number of occurrences $j$ and an exponent $m+1$) are defined in my Kempner paper for $j=0$. The notation is absent from my Irwin paper, due to condensed notations in the main Theorems. They are natural to consider in view of implementations and are fully documented via code comments to be found inside irwin_v5.sage.

  It is more difficult to apply parallelization to the computation of the recurrent coefficients u_{k;m} (or v_{k;m} for the series with positive coefficients).

  A variable maxworkers (defaulting to 8) configures how many cores the code will try to use, via the @parallel decorator.

  [!warning] The @parallel issue Sadly, an annoying issue has arisen with the use of @parallel, on the author recent Apple Silicon M4 hardware using macOS Sequoia. It is described in ParallelIssue.

  Keep in mind any change to maxworkers must be followed by a re-load of irwin_v5.sage in the interactive session.

  maxworkers does not have to be at most the actual number of cores on the user system. We tested both maxworkers=2 and maxworkers=8 on an old hardware with only 2 cores, timings were about the same.

  The memory footprint is higher than in v3 because maxworkers rows of the Pascal triangle are in memory rather than only 2 for v3.

  [!note] The number Mmax of terms to use from the Burnol series could be determined by the algorithm on an empirical basis, but due to inheritance from the original code written in 2012 by the author using Maple, its choice is made a priori, see irwin_v5_doc.pdf for the mathematical basis of this choice. When the excluded digit $d$ is $b-1$, Mmax could be chosen somewhat smaller, leading to some efficiency improvement, but it was chosen to keep the code simpler and not treat such case separately. Besides, with the 2025 versions, the higher terms are computed with much smaller precision than the main terms so that although they are more costly to compute theoretically, in practice dropping them would bring limited gain.

  Here are some timings of irwin() from the irwin_v5 module, on a Mac Mini M4 Pro with 10+4 cores:

  • version 1.5.4 of April 23, 2025, computed 2+101010 decimals of the "no-9" Kempner constant in 5h38mn, with maxworkers left to its default 8.
  • version 1.5.5 of April 30, 2025, computed 2+130010 decimals in 10h10mn, using maxworkers=10.

• Files with names of the type k_prec_2+N contain decimal expansions of the classic "no-9 radix-10" Kempner series 22.92067661926415..., correctly rounded to N decimal places.

  • k_prec_2+1000
  • k_prec_2+2000
  • k_prec_2+5000
  • k_prec_2+10000
  • k_prec_2+20000
  • k_prec_2+25000
  • k_prec_2+50000
  • k_prec_2+99998 (contributed by Yusuf Emin Akpınar, thanks!)

• taille_pascal.pdf explains how many bits are needed to store in computer memory the Pascal triangle up (or rather down) to a certain row, or the memory needed to store only one such row. Thanks to Nicolas Radulesco who asked for SageMath equivalents of the few Maple usages which were inserted in the text, they are in taille_pascal_symbolic.

Bibliographical references

This repository is devoted to the formulas now published in:

• Moments in the exact summation of the curious series of Kempner type. Amer. Math. Monthly 132:10 (2025), 995--1006. DOI.
• Measures for the summation of Irwin series. Integers 26:A11 (2026), 20pp. DOI.

Earlier numerical works computing otherwise Kempner and Irwin sums include:

• Robert Baillie: Sums of reciprocals of integers missing a given digit. Amer. Math. Monthly 86(5), 372–374 (1979) DOI
• Robert Baillie: Summing the curious series of Kempner and Irwin (2008). arXiv:0806.4410
• Thomas Schmelzer and Robert Baillie: Summing a curious, slowly convergent series. Amer. Math. Monthly 115(6), 525–540 (2008) DOI

More research by the author:

• Summing the "exactly one 42" and similar subsums of the harmonic series, Advances in Applied Mathematics Volume 162, January 2025, 102791. DOI.
• Sur l'asymptotique des sommes de Kempner pour de grandes bases, Publications Mathématiques de Besançon (2025 or 2026), to appear. Preprint available at arXiv:2403.01957.
• Digamma function and general Fischer series in the theory of Kempner sums, Expositiones Mathematicae, Volume 42, Issue 6, December 2024, 125604. DOI.
• Un développement asymptotique des sommes harmoniques de Kempner-Irwin, arXiv:2404.13763. Submitted.
• Measures associated with certain ellipsephic harmonic series and the Allouche-Hu-Morin limit theorem, Acta Mathematica Hungarica (2025) DOI.

Thanks

The February 2024 original version or irwin.sage which accompanied Measures for the summation of Irwin series was a revival and extension of Maple coding which I had done in 2012 and which regarded the results later described in Moments in the exact summation of the curious series of Kempner type (The American Mathematical Monthly 132:10 (2025), 995-1006). Preprint version at arXiv:2402.08525.

Thanks to Arnaud Bodin and Yusuf Emin Akpınar for sharing their interest during 2025 into the results of these two papers, and in particular for reporting progress in computing tens of thousands of digits of the classical "no-9" Kempner sum (Y. E. A. had obtained 100000=2+99998 digits using his C implementation of the formulas from my paper). This motivated me to revisit the February 2024 SageMath code to improve its usability for 10000 or more digits.

The changes allowed indeed to significantly improve computation times when asking for 10000+ digits. I then spent an inordinate amount of time about issue #1, and the sad conclusion I reached is that Python/SageMath are not suitable for parallelism on Apple's macOS and newer Macs.

License

The files in this repository are distributed under the CC-BY-SA 4.0 License. See LICENSE.