Plectis

Long paper

The Binary Totient Series

Erdős #249 By July 2026; revised 1 October 2026 Authorship, AI use and citation
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Summary. The dyadic totient sections through level e≥1 have an explicit rational basis of size 2e+1. Truncating a scaled tail difference produces an integer whose residue certifies nonintegrality when its distances from both endpoints of the residue interval exceed an explicit error bound. The record develops the resulting finite denominator exclusions, equivalent residue criteria, and sufficient exponential-sum estimates. Rational comparison sequences identify the limitations of particular size, parity, rank, and finite-prefix arguments. The short paper supplies the all-base basis theorem; here each conditional result carries the arithmetic hypothesis still needed to apply it. Those hypotheses are not established in the form required to prove irrationality of S=∑ φ(n)2−n.

This paper covers a claim-bounded problem-specific reasoning surface for Erdős #249: selected checked premises, finite certificates, closed routes, surviving obligations, and their exact evidence bands.

It does not establish the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #249, which remains open.

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References

  1. J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoret. Comput. Sci. 98 (1992), no. 2, 163–197, doi:10.1016/0304-3975(92)90001-V. Definitions and early statements are cited with the author preprint’s numbering.

  2. M. Coons, (Non)Automaticity of number theoretic functions, J. Théor. Nombres Bordeaux 22 (2010), no. 2, 339–352, doi:10.5802/jtnb.718. Theorem 3.2, p. 348.

  3. M. Coons, Regular sequences and the joint spectral radius, Internat. J. Found. Comput. Sci. 28 (2017), no. 2, 135–140, doi:10.1142/S0129054117500095; arXiv:1511.07535v1. Theorem 1, Proposition 4 and Corollary 7 are cited with the arXiv v1 numbering.

  4. G. Martin, Simultaneous inequalities among values of the Euler phi-function, arXiv:math/0603053v1, 2006. Theorem 1, pp. 1–2.

  5. E. Wong, answer 1211557 to An infinite sum based on the mod-parity of Euler’s totient function, Mathematics Stack Exchange, 29 March 2015, accessed 15 September 2026.

  6. P. Erdős, On the irrationality of certain series, Indag. Math. 19 (1957), 212–219, https://users.renyi.hu/~p_erdos/1957-07.pdf. Lemma 1, p. 213; Lemma 4, pp. 215–218; Lemma 4′, p. 218.

  7. H. Kaneko, Y. Suzuki and Y. Tachiya, Refinements of Erdős’s irrationality criterion for certain sparse infinite series, arXiv:2601.20743v1, 2026. Theorem A, p. 2; Theorem 3, p. 5; proof of Theorem A, p. 18.

  8. A. Eldar, comment and formula added to OEIS A256936 (revisions 28 and 31), 15 March 2026, https://oeis.org/history?seq=A256936, accessed 16 September 2026.

  9. S. Fan, comment on Erdős Problem #249, 16 May 2026, 19:01, Erdős Problems discussion thread, accessed 16 September 2026.

  10. P. Erdős and R. L. Graham, Old and New Problems and Results in Combinatorial Number Theory, Monographies de L’Enseignement Mathématique 28, 1980, p. 61.

  11. P. Erdős, On arithmetical properties of Lambert series, J. Indian Math. Soc. (N.S.) 12 (1948), 63–66, https://users.renyi.hu/~p_erdos/1948-04.pdf.

  12. K. Postelmans and W. Van Assche, Irrationality of ζq(1) and ζq(2), J. Number Theory 126 (2007), no. 1, 119–154, doi:10.1016/j.jnt.2006.11.011; arXiv:math/0604312v1. Theorem 1.3 is cited with the arXiv v1 pagination, p. 3.

  13. Yu. V. Nesterenko, Modular functions and transcendence questions, Sb. Math. 187 (1996), no. 9, 1319–1348, doi:10.1070/SM1996v187n09ABEH000158. Corollary 2, p. 1320.

  14. W. Schramm, The Fourier transform of functions of the greatest common divisor, Integers 8 (2008), #A50, https://math.colgate.edu/~integers/i50/i50.pdf. Theorem and equation (2), p. 2.

  15. L. Tóth, A survey of gcd-sum functions, J. Integer Seq. 13 (2010), Article 10.8.1, https://cs.uwaterloo.ca/journals/JIS/VOL13/Toth/toth10.pdf. §1, equations (1)–(2).

  16. M. Merca and M. D. Schmidt, Generating special arithmetic functions by Lambert series factorizations, Contrib. Discrete Math. 14 (2019), no. 1, 31–45, doi:10.55016/ojs/cdm.v14i1.62425; arXiv:1706.00393v2. Equation (1) is cited with the arXiv v2 pagination, p. 2.

  17. M. Merca, The Lambert series factorization theorem, Ramanujan J. 44 (2017), 417–435, doi:10.1007/s11139-016-9856-3. Theorem 1.2, p. 420.

  18. V. Kovač and T. Tao, On several irrationality problems for Ahmes series, Acta Math. Hungar. 175 (2025), no. 2, 572–608, doi:10.1007/s10474-025-01528-0; arXiv:2406.17593v4. Theorem 2.3 is cited with the arXiv v4 pagination, p. 5.

  19. R. Balasubramanian, S. Giri and P. Srivastav, On correlations of certain multiplicative functions, J. Number Theory 174 (2017), 221–238, doi:10.1016/j.jnt.2016.10.001; arXiv:1511.02221v3. Theorem 2.2 is cited with the arXiv v3 pagination, p. 2.

  20. A. Granville, Smooth numbers: computational number theory and beyond, in Algorithmic Number Theory, MSRI Publ. 44 (2008), 267–323, https://library.slmath.org/books/Book44/files/09andrew.pdf. Equations (1.1) and (1.3), p. 268.

  21. I. Schoenberg, Über die asymptotische Verteilung reeller Zahlen mod 1, Math. Z. 28 (1928), 171–199, doi:10.1007/BF01181156. §17, p. 193, for the distribution of φ(n)/n.

  22. I. J. Schoenberg, On asymptotic distributions of arithmetical functions, Trans. Amer. Math. Soc. 39 (1936), no. 2, 315–330. Theorem 1, pp. 318–319; §8, p. 323.

  23. N. Edeko, H. Kreidler and R. Nagel, A dynamical proof of the van der Corput inequality, Dyn. Syst. 37 (2022), no. 4, 648–665, doi:10.1080/14689367.2022.2100244; arXiv:2106.11835v3. Theorem 2.1 is cited with the arXiv v3 pagination, p. 4.

  24. K. Matomäki and M. Radziwiłł, Multiplicative functions in short intervals, Ann. of Math. 183 (2016), no. 3, 1015–1056, doi:10.4007/annals.2016.183.3.6; arXiv:1501.04585v4. Theorem 1 is cited with the arXiv v4 pagination, pp. 1–2.

  25. T. Tao, The logarithmically averaged Chowla and Elliott conjectures for two-point correlations, Forum Math. Pi 4 (2016), e8, doi:10.1017/fmp.2016.6; arXiv:1509.05422v4. Theorems 1.2–1.3 are cited with the arXiv v4 pagination, pp. 2 and 5.

  26. T. Tao and J. Teräväinen, Odd order cases of the logarithmically averaged Chowla conjecture, J. Théor. Nombres Bordeaux 30 (2018), no. 3, 997–1015, doi:10.5802/jtnb.1062; arXiv:1710.02112v1. Theorem 1.1 is cited with the arXiv v1 pagination, p. 2.

  27. B. Adamczewski and Y. Bugeaud, On the complexity of algebraic numbers I. Expansions in integer bases, Ann. of Math. 165 (2007), no. 2, 547–565. Theorem 1, p. 549.

  28. G. Everest, A. J. van der Poorten, Y. Puri and T. Ward, Integer sequences and periodic points, J. Integer Seq. 5 (2002), Article 02.2.3. Lemma 3.1 is cited with the preprint pagination, p. 5.

  29. P. Erdős and I. S. Gál, On the law of the iterated logarithm. I, Proc. Kon. Ned. Akad. Wetensch. Ser. A 58 = Indag. Math. 17 (1955), 65–76, https://www.renyi.hu/~p_erdos/1955-06.pdf.

  30. S. Yazdani, Multiplicative functions and k-automatic sequences, J. Théor. Nombres Bordeaux 13 (2001), no. 2, 651–658, https://www.numdam.org/item/JTNB_2001__13_2_651_0/. Theorem 2 and its proof, pp. 652–653; Corollary 4, p. 654.

  31. J.-P. Allouche, J. Shallit and R. Yassawi, How to prove that a sequence is not automatic, Exposition. Math. 40 (2022), no. 1, 1–22, doi:10.1016/j.exmath.2021.08.001; arXiv:2104.13072v1. Theorem and example numbering follows the arXiv v1 (2021).

  32. J. Bell and D. Smertnig, Mahler series with multiplicative coefficient sequences, arXiv:2603.23456v1, 24 March 2026, preprint. Theorem 1.3, p. 2; Lemma 3.3, p. 9.

  33. J. P. Bell, N. Bruin and M. Coons, Transcendence of generating functions whose coefficients are multiplicative, Trans. Amer. Math. Soc. 364 (2012), no. 2, 933–959, doi:10.1090/S0002-9947-2011-05479-6; arXiv:1003.2221v2. Theorems 1.5–1.6 are cited with the arXiv v2 pagination, p. 2.

  34. B. Adamczewski, M. Drmota and C. Müllner, (Logarithmic) densities for automatic sequences along primes and squares, Trans. Amer. Math. Soc. 375 (2022), no. 1, 455–499, doi:10.1090/tran/8476; arXiv:2009.14773v2. Theorem 1.4 and Remark 1.5, p. 4, and the proof at the end of Section 7, p. 19, refer to the arXiv v2 (2021).

  35. B. Adamczewski, J. Bell and D. Smertnig, A height gap theorem for coefficients of Mahler functions, J. Eur. Math. Soc. 25 (2023), no. 7, 2525–2571, doi:10.4171/JEMS/1244. Theorem 1.2(b), p. 2528; logarithmic height in Section 1.1.1, p. 2527.

  36. T. Crmarić and V. Kovač, On the irrationality of certain super-polynomially decaying series, arXiv:2504.18712v1 (2025), Lemma 4(a) and Remark 5.

  37. M. Coons, (Non)Automaticity of number theoretic functions, arXiv:0810.3709v3, Theorem 3.3. This version’s numbering differs from the journal version cited above.

  38. K. Habiro, Cyclotomic completions of polynomial rings, Publ. Res. Inst. Math. Sci. 40 (2004), 1127–1146, Theorem 5.2, p. 1138.

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Authorship and AI use. Will Cook built and directed the research infrastructure and maintains the public release. He reviewed claims when he could. AI agents did most of the research and drafting. Cook did not independently verify every claim.

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