block-Irwin sums

This repository will gather several methods for computing to arbitrary precision the block-Irwin sums

$$I(b,w,k)=\sum_{n>0,\ k_w(n)=k}\frac1n ,$$

where $b>1$ is the base, $w$ a block of base-$b$ digits and $k_w(n)$ the number of occurrences of $w$ in the base-$b$ representation of $n$, counted with overlaps.

References:

• J.-F. Burnol, Block-count constrained harmonic sums: spectral expansion and block-directed Euler–Maclaurin, arXiv:2609.17045.
• J.-F. Burnol, The modal expansion of Kempner sums, arXiv:2609.31986.

Methods:

1. Script blockirwin.py — full documentation: README-method1.md. Proposition 11 of arXiv:2609.17045 expresses $I(b,w,k)$ through Stieltjes values of the measures $\eta_j$, Proposition 12 (level raising) pushes their arguments to at least $\ell$ digits, and these values are computed as alternating series in the moments of the $\eta_j$, obtained from the finite triangular recurrences of Proposition 13. Any base, any block, any $k\ge0$.
2. Script modalirwin.py - full documentation: README-modalirwin.md. Implements the spectral/modal series of arXiv:2609.17045, Part 2 ($k\ge1$), and the modal Kempner series of arXiv:2609.31986 ($k=0$). It uses the published triangular recurrences for the moments of $\eta_0$, $\mu_0$, and the tilted measures $\nu_j$, and the corresponding modal coefficients, with level-raised Newton series for the auxiliary Stieltjes integrals.
3. Uncertain, may never be done: modal expansions applied once a certain level has been reached.

Performance and dependence on k. blockirwin.py (method 1) becomes more costly as k grows. In contrast, modalirwin.py becomes faster with increasing k. For example, at b=10, w="314", 40 digits (best of three runs, October 12, 2026, CPython 3.13.16, mpmath 1.4.1), calls to blockirwin.py took 0.026 s (k=1) versus 0.039 s (k=10), while modalirwin.py 0.6.0 took 0.305 s versus 0.0071 s. Of course timings are heavily hardware dependent. This is an illustration, not a universal ordering: method 1 often wins (by orders of magnitude). For details about the heuristically estimated cubic cost in the number of modes see README-modalirwin.md.

Usage and validation

from blockirwin import blockirwin
from modalirwin import modalirwin

print(blockirwin(2, "01", 0, 20))
print(modalirwin(2, "01", 0, 20))
modalirwin(2, "01", 2, 20, modalcoeffs=True)

python compare_modalirwin.py 2 01 0 20
python -m pytest -q test_blockirwin.py test_modalirwin.py test_modalirwin_cancellation.py

The modules use Python's standard library and mpmath. See some comments about mpmath version at bottom of README-method1.md.

AI assistance

The initial blockirwin.py, test_blockirwin.py, README-method1.md, and this README.md's original version were produced on September 28–29, 2026 by Claude Opus 5.5 and Claude Sonnet 5 (Anthropic), from the mathematics and earlier Maple/SageMath/mpmath code of Jean-François Burnol.

The complementary modalirwin.py, and other files inclusive of test_modalirwin.py and compare_modalirwin.py were produced with OpenAI ChatGPT (GPT-6) on October 10-11, 2026, from the two Burnol preprints.

This README.md was extended by ChatGPT (GPT-6) to cover this, and then edited (twice: on October 10 and again on October 11) by a human, to prune some too lengthy (and too heavily decorated like this) ChatGPT outputs.

On October 12, 2026, Claude Opus 5.5 (Anthropic) made test_blockirwin.py run fully under current pytest, and revised the ChatGPT modalirwin.py into version 0.6.0: exact integer level raising built once per call, removal of a defect which forced full precision in all modes, and new default levels; one test of test_modalirwin.py and README-modalirwin.md were adapted accordingly.

The mathematics is fully authored by Jean-François Burnol. It was parsed by ChatGPT 5.6 Sol to search for errors (and improve the English language use).

The repository is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License.