Reciprocal Mersenne Subseries

Will Cook

Long paper

Reciprocal Mersenne Subseries

Erdős #257 By Will Cook 30 September 2026 Authorship, AI use and citation

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Summary. A weighted summability criterion proves irrationality of reciprocal Mersenne subseries: if h is the part of a supported on a finite nonempty prime set P, every infinite A with ∑ $h (a) / (a (b^{h (a)} - 1))$ finite has irrational ∑ $(b^{a} - 1)^{- 1}$ . The base-two condition gives the conclusion at every integer base for every infinite subset, and permits an example with divergent reciprocal sum, while excluding full support and all odd exponents. The proof averages over a finite part of the support and over dyadic observation lengths. Erdős's weaker reciprocal-summable theorem is proved in full, and a common window extends both arguments to unions of weighted and positive-cover supports. Finite-denominator and achievement-set calculations formulate tests for the targets $1 / 2$ and $1 / 21$ ; those tests do not decide either target or the universal question. Prime support at base 2 and squarefree support at power-of-two bases are attributed as existing work.

This paper covers a claim-bounded problem-specific reasoning surface for Erdős #257: selected checked premises, finite computations, 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 #257, whose universal and half-value questions remain open.

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References

P. Erdős, On arithmetical properties of Lambert series, J. Indian Math. Soc. 12 (1948), 63–66.
P. Erdős, On the irrationality of certain series, Math. Student 36 (1968), 222–226 (issued 1969); five-page scan. The theorem on p. 222 treats pairwise-coprime support with convergent reciprocal sum at every integer base $b \geq 2$ ; the claimed removal of pairwise coprimality is stated without proof. The same page says the reciprocal-sum condition could be replaced by a weaker but more complicated condition, and p. 226 suggests $\sum_{n_{i} < x} 1 / n_{i} = o (\log \log x)$ for pairwise coprime supports.
F. Luca and Y. Tachiya, Linear independence results for the values of divisor functions series, RIMS Kôkyûroku No. 2014 (2017), 138–150. Theorem A (p. 139) restates Theorem 1.1 of their paper Irrationality of Lambert series associated with a periodic sequence, Int. J. Number Theory 10 (2014), no. 3, 623–636, DOI: for a purely periodic integer sequence $a (n)$ , not identically zero, $\sum_{n \geq 1} a (n) / (q^{n} - 1)$ is irrational for every integer $q$ with $| q | > 1$ . Example 2 (p. 140) treats the odd support.
Z. Nitecki, Subsum sets: intervals, Cantor sets, and Cantorvals, arXiv:1106.3779v2 (2013). Theorem 4 (p. 9) shows that when every term exceeds its tail the subsum set is a Cantor set of Lebesgue measure $lim_{n} 2^{n} X_{n}$ , where $X_{n}$ is the $n$ th tail, and credits this to H. Hornich (1941).
N. Sedaghat, T. Stephen, and L. Chindelevitch, Speeding up dualization in the Fredman–Khachiyan Algorithm B, 17th International Symposium on Experimental Algorithms (SEA 2018), LIPIcs 103 (2018), 6:1–6:13.
H. Kaneko, Y. Suzuki, and Y. Tachiya, Refinements of Erdős’s irrationality criterion for certain sparse infinite series, arXiv:2601.20743v1 (2026). The averaged tail $R_{c} (q, x, z)$ , a sum over offsets $j \geq z$ , is (1.7) and Theorem 1 is on p. 3; its rational-integer form, Theorem 3, is on p. 5. Lemma 1 (pp. 6–7) derives irrationality from arbitrarily small nonzero scaled tails, and Lemma 2 (pp. 7–8) shows that most scaled tails in a range are small under a counting condition on the support. Their criteria assume sparse coefficient supports.
D. Duverney and Y. Tachiya, Refinement of the Chowla–Erdős method and linear independence of certain Lambert series, Forum Math. 31 (2019), no. 6, 1557–1566, DOI. Page numbers refer to the linked author preprint. Theorem 1.1 (pp. 2–3) is the refined Chowla–Erdős criterion; its proof in Section 2 (pp. 5–7) selects an index by minimising a local coefficient mass along an arithmetic progression, (2.3)–(2.9). Corollary 1.2 gives the general $F_{s} (E)$ theorem and its monomial images under $| q | lcm (1, \dots, ℓ) \leq s$ , and at every integer base when $s = \infty$ ; Example 1.1 gives the joint squarefree family at all bases $2^{j}$ , $j \geq 1$ . Both are on p. 4, which also relates Corollary 1.2 to the conjecture of Erdős and Graham; the proof of Corollary 1.2 is on pp. 10–11.
T. Tao and J. Teräväinen, Quantitative correlations and some problems on prime factors of consecutive integers, arXiv:2512.01739v2 (submitted December 2025, revised April 2026). Theorem 1.3 (p. 4) proves $\sum_{n \geq 1} ω (n) / 2^{n} = \sum_{p} (2^{p} - 1)^{- 1}$ irrational, settling the prime-support case of #257 at base $2$ ; the paragraph after it states that the prime-support result extends to every integer base $b \geq 2$ , and separately that full prime-power support is irrational at base $2$ . The modifications for those two assertions are left to the reader; arbitrary subsets of the primes or prime powers are not treated there. Theorem 3.1 is on p. 24, and the proof of Theorem 1.3 is Section 5, pp. 44–56.
W. Zudilin, Heine’s basic transform and a permutation group for $q$ -harmonic series, Acta Arith. 111 (2004), no. 2, 153–164, DOI. Theorem 1 (p. 154) gives $μ (h_{p} (1)) \leq 2.46497868 \dots$ for $h_{p} (1) = \sum_{n \geq 1} 1 / (p^{n} - 1)$ , uniformly in the integer $p$ with $| p | \geq 2$ ; the remark at the end of Section 3 (p. 159) corrects the computation of the author’s earlier paper.
J. M. Campbell, On the binary digits of the Erdős–Borwein constant, arXiv:2605.24160v1 (2026). Theorem 1 (p. 12) proves that the block 11 occurs infinitely often in the binary expansion of $\sum_{n \geq 1} 1 / (2^{n} - 1)$ , answering a question of Crandall recalled in the introduction.
K. Matomäki and M. Radziwiłł, Multiplicative functions in short intervals, Ann. of Math. (2) 183 (2016), no. 3, 1015–1056, DOI. Page numbers refer to arXiv:1501.04585v4.
A. P. Mangerel, Divisor-bounded multiplicative functions in short intervals, Res. Math. Sci. 10 (2023), no. 1, Paper No. 12, DOI. Page numbers refer to arXiv:2108.11401v2.
Y.-C. Sun, The critical-window profile for the $k$ -fold divisor function in almost all short intervals, arXiv:2401.08432v3 (2026).
F. Luca and Y. Tachiya, Irrationality of Lambert series associated with a periodic sequence, International Journal of Number Theory 10 (2014), no. 3, 623–636. doi:10.1142/S1793042113501121.
F. Luca and Y. Tachiya, Linear independence of certain Lambert series, Proceedings of the American Mathematical Society 142 (2014), no. 10, 3411–3419. doi:10.1090/S0002-9939-2014-12102-2.
W. Van Assche, Little $q$ -Legendre polynomials and irrationality of certain Lambert series, The Ramanujan Journal 5 (2001), 295–310. doi:10.1023/A:1012930828917; arXiv:math/0101187v1.
H. Hornich, Über beliebige Teilsummen absolut konvergenter Reihen, Monatshefte für Mathematik und Physik 49 (1941), 316–320. doi:10.1007/BF01707309.
S. Głąb and F. Prus-Wiśniowski, Achievement sets – current results and open problems, arXiv:2512.17285v1 (2025), preprint. https://arxiv.org/abs/2512.17285v1.
V. Kovač and T. Tao, On several irrationality problems for Ahmes series, Acta Math. Hungar. 175 (2025), no. 2, 572–608, DOI. Page numbers refer to arXiv:2406.17593v4. Remark 4.1 (p. 13) records the strict tail inequality and the Cantor structure. Theorem 2.3 (p. 5; proof pp. 13–14) constructs rational merged sums from several bases under its mass hypothesis; the construction merges several bases and leaves fixed-base #257 untouched.
W. van Doorn and V. Kovač, Lacunary sequences whose reciprocal sums represent all rational numbers in an interval, Acta Arith. 223 (2026), 275–295, DOI. The theorem and page references used here refer to arXiv:2509.24971v3 (3 December 2025), https://arxiv.org/abs/2509.24971v3. Theorem 1 is on p. 2; Lemma 4 and Corollary 5 are on pp. 5–7.
P. Erdős, Some problems and results on the irrationality of the sum of infinite series, J. Math. Sci. 10 (1975), 1–7.
K. Barreto, J. Kang, S.-H. Kim, V. Kovač, and S. Zhang, Irrationality of rapidly converging series: a problem of Erdős and Graham, arXiv:2601.21442v3 (2026), to appear in Bull. London Math. Soc.
K. Conrad, The Chinese remainder theorem, expository notes, Theorem 3.1 (pp. 2–3). The compatibility criterion for noncoprime moduli used above is obtained by the displayed prime-power reduction.
P. Erdős, Beweis eines Satzes von Tschebyschef, 1932, pp. 194–198, especially §§4–5.

About this paper

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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[ref-1] P. Erdős, On arithmetical properties of Lambert series, J. Indian Math. Soc. 12 (1948), 63–66.

[ref-2] P. Erdős, On the irrationality of certain series, Math. Student 36 (1968), 222–226 (issued 1969); five-page scan. The theorem on p. 222 treats pairwise-coprime support with convergent reciprocal sum at every integer base $b \geq 2$ ; the claimed removal of pairwise coprimality is stated without proof. The same page says the reciprocal-sum condition could be replaced by a weaker but more complicated condition, and p. 226 suggests $\sum_{n_{i} < x} 1 / n_{i} = o (\log \log x)$ for pairwise coprime supports.

[ref-3] F. Luca and Y. Tachiya, Linear independence results for the values of divisor functions series, RIMS Kôkyûroku No. 2014 (2017), 138–150. Theorem A (p. 139) restates Theorem 1.1 of their paper Irrationality of Lambert series associated with a periodic sequence, Int. J. Number Theory 10 (2014), no. 3, 623–636, DOI: for a purely periodic integer sequence $a (n)$ , not identically zero, $\sum_{n \geq 1} a (n) / (q^{n} - 1)$ is irrational for every integer $q$ with $| q | > 1$ . Example 2 (p. 140) treats the odd support.

[ref-4] Z. Nitecki, Subsum sets: intervals, Cantor sets, and Cantorvals, arXiv:1106.3779v2 (2013). Theorem 4 (p. 9) shows that when every term exceeds its tail the subsum set is a Cantor set of Lebesgue measure $lim_{n} 2^{n} X_{n}$ , where $X_{n}$ is the $n$ th tail, and credits this to H. Hornich (1941).

[ref-5] N. Sedaghat, T. Stephen, and L. Chindelevitch, Speeding up dualization in the Fredman–Khachiyan Algorithm B, 17th International Symposium on Experimental Algorithms (SEA 2018), LIPIcs 103 (2018), 6:1–6:13.

[ref-6] H. Kaneko, Y. Suzuki, and Y. Tachiya, Refinements of Erdős’s irrationality criterion for certain sparse infinite series, arXiv:2601.20743v1 (2026). The averaged tail $R_{c} (q, x, z)$ , a sum over offsets $j \geq z$ , is (1.7) and Theorem 1 is on p. 3; its rational-integer form, Theorem 3, is on p. 5. Lemma 1 (pp. 6–7) derives irrationality from arbitrarily small nonzero scaled tails, and Lemma 2 (pp. 7–8) shows that most scaled tails in a range are small under a counting condition on the support. Their criteria assume sparse coefficient supports.

[ref-7] D. Duverney and Y. Tachiya, Refinement of the Chowla–Erdős method and linear independence of certain Lambert series, Forum Math. 31 (2019), no. 6, 1557–1566, DOI. Page numbers refer to the linked author preprint. Theorem 1.1 (pp. 2–3) is the refined Chowla–Erdős criterion; its proof in Section 2 (pp. 5–7) selects an index by minimising a local coefficient mass along an arithmetic progression, (2.3)–(2.9). Corollary 1.2 gives the general $F_{s} (E)$ theorem and its monomial images under $| q | lcm (1, \dots, ℓ) \leq s$ , and at every integer base when $s = \infty$ ; Example 1.1 gives the joint squarefree family at all bases $2^{j}$ , $j \geq 1$ . Both are on p. 4, which also relates Corollary 1.2 to the conjecture of Erdős and Graham; the proof of Corollary 1.2 is on pp. 10–11.

[ref-8] T. Tao and J. Teräväinen, Quantitative correlations and some problems on prime factors of consecutive integers, arXiv:2512.01739v2 (submitted December 2025, revised April 2026). Theorem 1.3 (p. 4) proves $\sum_{n \geq 1} ω (n) / 2^{n} = \sum_{p} (2^{p} - 1)^{- 1}$ irrational, settling the prime-support case of #257 at base $2$ ; the paragraph after it states that the prime-support result extends to every integer base $b \geq 2$ , and separately that full prime-power support is irrational at base $2$ . The modifications for those two assertions are left to the reader; arbitrary subsets of the primes or prime powers are not treated there. Theorem 3.1 is on p. 24, and the proof of Theorem 1.3 is Section 5, pp. 44–56.

[ref-9] W. Zudilin, Heine’s basic transform and a permutation group for $q$ -harmonic series, Acta Arith. 111 (2004), no. 2, 153–164, DOI. Theorem 1 (p. 154) gives $μ (h_{p} (1)) \leq 2.46497868 \dots$ for $h_{p} (1) = \sum_{n \geq 1} 1 / (p^{n} - 1)$ , uniformly in the integer $p$ with $| p | \geq 2$ ; the remark at the end of Section 3 (p. 159) corrects the computation of the author’s earlier paper.

[ref-10] J. M. Campbell, On the binary digits of the Erdős–Borwein constant, arXiv:2605.24160v1 (2026). Theorem 1 (p. 12) proves that the block 11 occurs infinitely often in the binary expansion of $\sum_{n \geq 1} 1 / (2^{n} - 1)$ , answering a question of Crandall recalled in the introduction.

[ref-11] K. Matomäki and M. Radziwiłł, Multiplicative functions in short intervals, Ann. of Math. (2) 183 (2016), no. 3, 1015–1056, DOI. Page numbers refer to arXiv:1501.04585v4.

[ref-12] A. P. Mangerel, Divisor-bounded multiplicative functions in short intervals, Res. Math. Sci. 10 (2023), no. 1, Paper No. 12, DOI. Page numbers refer to arXiv:2108.11401v2.

[ref-13] Y.-C. Sun, The critical-window profile for the $k$ -fold divisor function in almost all short intervals, arXiv:2401.08432v3 (2026).

[ref-14] F. Luca and Y. Tachiya, Irrationality of Lambert series associated with a periodic sequence, International Journal of Number Theory 10 (2014), no. 3, 623–636. doi:10.1142/S1793042113501121.

[ref-15] F. Luca and Y. Tachiya, Linear independence of certain Lambert series, Proceedings of the American Mathematical Society 142 (2014), no. 10, 3411–3419. doi:10.1090/S0002-9939-2014-12102-2.

[ref-16] W. Van Assche, Little $q$ -Legendre polynomials and irrationality of certain Lambert series, The Ramanujan Journal 5 (2001), 295–310. doi:10.1023/A:1012930828917; arXiv:math/0101187v1.

[ref-17] H. Hornich, Über beliebige Teilsummen absolut konvergenter Reihen, Monatshefte für Mathematik und Physik 49 (1941), 316–320. doi:10.1007/BF01707309.

[ref-18] S. Głąb and F. Prus-Wiśniowski, Achievement sets – current results and open problems, arXiv:2512.17285v1 (2025), preprint. https://arxiv.org/abs/2512.17285v1.

[ref-19] V. Kovač and T. Tao, On several irrationality problems for Ahmes series, Acta Math. Hungar. 175 (2025), no. 2, 572–608, DOI. Page numbers refer to arXiv:2406.17593v4. Remark 4.1 (p. 13) records the strict tail inequality and the Cantor structure. Theorem 2.3 (p. 5; proof pp. 13–14) constructs rational merged sums from several bases under its mass hypothesis; the construction merges several bases and leaves fixed-base #257 untouched.

[ref-20] W. van Doorn and V. Kovač, Lacunary sequences whose reciprocal sums represent all rational numbers in an interval, Acta Arith. 223 (2026), 275–295, DOI. The theorem and page references used here refer to arXiv:2509.24971v3 (3 December 2025), https://arxiv.org/abs/2509.24971v3. Theorem 1 is on p. 2; Lemma 4 and Corollary 5 are on pp. 5–7.

[ref-21] P. Erdős, Some problems and results on the irrationality of the sum of infinite series, J. Math. Sci. 10 (1975), 1–7.

[ref-22] K. Barreto, J. Kang, S.-H. Kim, V. Kovač, and S. Zhang, Irrationality of rapidly converging series: a problem of Erdős and Graham, arXiv:2601.21442v3 (2026), to appear in Bull. London Math. Soc.

[ref-23] K. Conrad, The Chinese remainder theorem, expository notes, Theorem 3.1 (pp. 2–3). The compatibility criterion for noncoprime moduli used above is obtained by the displayed prime-power reduction.

[ref-24] P. Erdős, Beweis eines Satzes von Tschebyschef, 1932, pp. 194–198, especially §§4–5.