Higher-Order Congruence for Reciprocal Power Sums and Generalized Lehmer-Type Products

Hao Zhong; Zhenming Tang

Higher-Order Congruence for Reciprocal Power Sums and Generalized Lehmer-Type Products

T0 review · 0 major / 6 minor · reviewed 2026-07-14 · grok-4.5

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Pith's one-line read Odd-order reciprocal power sums over residues coprime to n are congruent modulo n to an explicit Bernoulli-polynomial expression, while generalized Lehmer products expand as truncated Bell polynomials modulo n^{K+1}.

desk verdict Solid incremental extension of the Cai–Zhong line: odd-order reciprocal sums via Bernoulli polynomials and a clean Bell packaging of higher Lehmer products. read the letter →

arxiv 2607.11113 v1 pith:JTXVVP2Q submitted 2026-07-13 math.NT

Zhenming Tang , Hao Zhong This is my paper · ORCID

classification math.NT MSC 11A0711B6811B73

keywords reciprocalpowersumsBernoullipolynomialsLehmer-typeproductscompleteexponentialBellhigher-ordercongruencesMöbiusinversionEulertotient

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper extends classical Morley–Cai–Zhong congruences for reciprocal squares and binomial products to two broader settings. First, for n coprime to 6 and e in {2,3,4,6}, it proves that the truncated sum of 1/r^m over residues r ≤ floor(n/e) coprime to n is congruent modulo n to a simple multiple of a difference of Bernoulli polynomials, provided m is an odd integer lying in a range controlled by the prime-power factors of n. Second, the Möbius product of binomial coefficients binom(kd-1, floor(d/e)) is shown to admit an explicit power-series expansion in n whose coefficients are complete exponential Bell polynomials evaluated at the reciprocal power sums; the expansion holds modulo n^{K+1} whenever n is coprime to K!. Together the two theorems supply a uniform, computable framework that recovers all previously known low-order cases and makes higher-order verification algorithmic.

What carries the argument

The complete exponential Bell polynomials that convert the truncated logarithm of the product ∏(1-kn/r) into an explicit power series in n, together with a Kummer-type congruence for Bernoulli polynomials that reduces the large local index M+1 to the global totient index φ(n)-m+1.

What would settle it

Pick a concrete prime power p^l (p≥5) and an admissible odd m, compute the truncated sum S_m(p^l) by direct enumeration and the right-hand side of Theorem 2.2 involving Bernoulli polynomials, and check whether they agree modulo p^l; any mismatch falsifies the local step on which the whole paper depends.

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Extended reading notes

Core claim

Under the arithmetic hypotheses (n,6)=1, e∈{2,3,4,6} and m odd with 3≤m≤min(φ(p^l)-l) and m≢1 mod(p-1) for every p^l∥n, the sum S_m(n) satisfies the uniform congruence modulo n given in Theorem 1.3; independently, for any K with (n,K!)=1 the generalized Lehmer product equals (-1)^{φ_e(n)} times the partial sum of Bell polynomials ar B_m(-k S_1,…,-(m-1)!k^m S_m)/m! · n^m modulo n^{K+1} (Theorem 1.4).

Load-bearing premise

The argument rests on a Kummer-type congruence that lets one replace a large Bernoulli-polynomial index by a smaller congruent index; if that replacement fails for the allowed range of m, both the local formula and its lift to composite n collapse.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This is a clean, self-contained next step in the Cai–Zhong program on Lehmer-type products and reciprocal power sums. Two concrete new statements: Theorem 1.3 gives a uniform mod-n congruence for the truncated odd-order sum Sm(n) under the usual (n,6)=1 and e in {2,3,4,6} hypotheses, expressed as an explicit multiple of a difference of Bernoulli-polynomial values; Theorem 1.4 packages the Möbius product of binomials as a truncated expansion in complete exponential Bell polynomials modulo n^{K+1}. Both go beyond the square-sum and cubic-product results of their earlier arXiv:1604.00445 and Cai’s series.

What they do well is keep the bookkeeping honest. Local prime-power congruences come from Euler reduction of inverses, Faulhaber plus the addition formula for Bernoulli polynomials, then a Kummer-type index drop (Lemma 2.2, cited from Ma–Li) whose hypotheses match exactly the range they enforce on m. The lift to composite n is by CRT after a short Möbius argument; the product side is just the truncated log-exp generating function that defines the Bell polynomials, with an optional recursive form for computation. No free parameters, no circular reductions, and the base cases from [14] are used only as motivation, not as black boxes that hide the new work.

Soft spots are minor and proportional. The significance is specialized—useful computational scaffolding inside a narrow arithmetic-congruence niche, not a method that travels far. The Kummer input is external, but it is published and the paper’s conditions sit inside its hypotheses, so the dependence is clean rather than load-bearing risk. Presentation is dense but readable if you already know the earlier papers.

This is for people already working on Bernoulli-polynomial congruences or algorithmic verification of Lehmer products. A serious referee will find the derivations standard and carefully tracked; it deserves peer review and should clear a specialized number-theory journal after routine polishing. I would cite the Bell expansion if I needed higher-order product formulae, and I would bring the odd-power statement to a reading group only if we were already deep in this literature.

Referee Report

0 major / 6 minor

Summary. The paper extends quadratic reciprocal-sum congruences of Zhong–Chern–Cai to odd powers: under (n,6)=1, e∈{2,3,4,6} and odd m in the range 3≤m≤min(φ(p^l)-l) with m≢1 (mod p-1) for all p|n, Theorem 1.3 asserts that the truncated sum S_m(n) is congruent modulo n to an explicit multiple (via a generalized Jordan totient) of the difference of two Bernoulli-polynomial values at 1/e. Independently, Theorem 1.4 supplies a truncated expansion of the Möbius product of binomial coefficients ∏_d|n binom(kd-1,⌊ d/e⌋)^μ(n/d) in complete exponential Bell polynomials, valid modulo n^{K+1} whenever (n,K!)=1; the K=2 case recovers the earlier cubic congruence. Both results are obtained by local prime-power analysis (Euler reduction of inverses, Faulhaber/Bernoulli expansions, Kummer-type index reduction) followed by CRT and Möbius inversion, with an explicit recursive form of the Bell coefficients provided for computation.

Significance. If the statements hold, the work supplies a uniform, algorithmically usable framework that lifts the classical Morley–Lehmer–Cai line from squares and cubic products to odd reciprocal powers and arbitrary modulus order. The Bell-polynomial expansion and its recursive companion make higher-order verification concrete rather than merely existential; the local-to-global passage via CRT and the careful matching of hypotheses with the cited Kummer congruence for Bernoulli polynomials are clean applications of standard tools. The results therefore constitute a genuine, if incremental, advance in the arithmetic of truncated harmonic sums and binomial products.

minor comments (6)

Throughout §§1–2 the displayed formulae for the generalized totient φ_f^{(k)}(n) and for the right-hand side of Theorem 1.3 are typographically broken (superscripts and subscripts collide, e.g. “n φ(n)-mφ(m-φ(n)) 1 (n)”). These must be restored to standard LaTeX so that the main statements are readable.
Lemma 2.2 (the Kummer-type congruence) is cited from Ma–Li arXiv:2211.15874; a one-sentence reminder of the precise range of indices and the coprimality condition on the argument would help the reader verify that the paper’s hypotheses on m exactly match the lemma’s requirements.
In the derivation of the lifting relation (33) the case c ≡ -1 (mod e) uses the identity 1/r^m ≡ (-1)^m / (pl-tr)^m; the subsequent sign (-1)^{m-1} is correct for odd m, but a parenthetical remark that the same identity fails for even m (hence the restriction) would prevent confusion.
The recursive definition (53) of the coefficients C_t is useful; it would be clearer if the authors explicitly noted that it is the standard Newton–Girard recurrence for the elementary-to-power-sum conversion underlying the Bell polynomials.
Several references appear only as arXiv preprints (e.g. [1],[4]–[10],[12]); where published versions exist they should be cited, and the arXiv numbers should be updated to the final versions if available.
Minor linguistic points: “high-order” o “higher-order” in the title and abstract; “we already know the form” is informal for a journal abstract; the phrase “certain reciprocal sums of odd order” should specify the precise arithmetic constraints already at the abstract level.

Circularity Check

1 steps flagged · score 2.0 of 10

Minor self-citation of base cases from overlapping authors; central odd-power and Bell expansions are derived independently from classical tools.

self citation load bearing [Introduction, Theorems 1.1–1.2 and surrounding text; recovery after Theorem 1.4]
"Zhong et al.[14] further derived quadratic reciprocal-sum congruences and product-type congruences... As a continuation of the work of Cai[3], Zhong et al.[14] proposed two main congruences below: Theorem 1.1([14], Theorem 1.1)... Theorem 1.2([14], Theorem 1.2)... For K=2, this congruence recovers the form obtained in [14]."

The square-sum congruence and the mod-n^3 product are taken from prior work whose author list overlaps the present paper (Hao Zhong). They are used only as motivational base cases and recovered as the K=2 specialization; the odd-m and higher-K statements are proved from independent classical machinery and do not logically depend on the truth of those earlier claims.

full rationale

The derivation chains for Theorems 1.3 and 1.4 are self-contained. Local prime-power congruences for Sm(pl) follow from Euler reductions of reciprocals, Faulhaber/Bernoulli expansions of power sums (Lemma 2.1 refining Cai et al.), a Kummer-type index reduction (external Lemma 2.2), a lifting relation proved in-paper (Lemma 2.3), and Sun’s congruence plus CRT for the global form. The product expansion is the truncated exp-log series of Tn rewritten via the definition of complete exponential Bell polynomials, followed by Möbius inversion relating Tn to the binomial product; the K=2 recovery is an immediate specialization, not an assumption. The only self-citations ([14]) supply motivational base cases (square sums and cubic products) that are not load-bearing for the new statements. No equation reduces tautologically to a fitted quantity, a definition of the target, or an unverified uniqueness claim. Score 2 reflects the minor overlapping-author citation of base cases while the novel content stands independently.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The work rests entirely on classical arithmetic tools and previously published congruences; no free parameters are fitted and no new mathematical objects are postulated. The only non-standard inputs are the domain restrictions (n coprime to 6, e in {2,3,4,6}, m odd and bounded by totients) that keep the Euler and Kummer machinery applicable.

assumptions (5)

domain assumption Kummer-type congruence for Bernoulli polynomials: BN(x)/N ≡ BN'(x)/N' mod p^l when N ≡ N' mod φ(p^l), N,N' even > l and p-1 does not divide them (Lemma 2.2).
Invoked to reduce the large index M+1 to φ(p^l)-m+1; taken from Ma–Li arXiv:2211.15874 without re-proof.
domain assumption Sun’s general congruence relating Bernoulli polynomials at rational arguments to truncated power sums (Lemma 2.4).
Used to equate local and global Bernoulli indices modulo p^l; cited from Discrete Math. 2003.
standard math Faulhaber’s formula expressing power sums via Bernoulli polynomials.
Applied in the expansion of ∑(p^l - t r)^M.
standard math Definition and generating function of complete exponential Bell polynomials.
Used to rewrite the truncated exponential of the logarithmic series of the product.
domain assumption (n,6)=1 and e ∈ {2,3,4,6} so that the Jacobi symbol Je(n) is well-defined and the least residues modulo e are only 1 or e-1.
Stated at the outset and used throughout to control signs and Bernoulli evaluations at 1/e.

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Cite this review

Pith. "Pith review of Higher-Order Congruence for Reciprocal Power Sums and Generalized Lehmer-Type Products." pith.science (2026). https://pith.science/paper/JTXVVP2Q

@misc{pith2026260711113,
  author       = {Pith},
  title        = {Pith review of: Higher-Order Congruence for Reciprocal Power Sums and Generalized Lehmer-Type Products},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JTXVVP2Q}},
  note         = {Machine review of arXiv:2607.11113}
}

abstract

This paper investigates high-order congruences of reciprocal power sums and Lehmer-type products. Let $n\geq 1$ with $(n,6)=1$ and $e\in\{2,3,4,6\}$. For the reciprocal square sums \begin{equation*} S(n)=\sum_{\substack{r=1 \\ (r,n)=1}}^{\lfloor n/e \rfloor}\frac{1}{r^2} \end{equation*} we already know the form of the congruence modulo $n$. In this paper, motivated by the known congruences, we first extend these results to certain reciprocal sums of odd order and establish a uniform congruence modulo $n$ for \begin{equation*} S_m(n)=\sum_{\substack{r=1 \\ (r,n)=1}}^{\lfloor n/e \rfloor}\frac{1}{r^m} \end{equation*} We then study the generalized Lehmer-type product \begin{equation*} \prod_{d \mid n}\binom{kd-1}{\lfloor d/e \rfloor}^{\mu(n/d)} \end{equation*} Although congruences modulo $n^3$ for this product have previously been obtained, higher-order congruences do not admit a comparably simple closed form. To address this difficulty, we derive an explicit truncated expansion in terms of complete exponential Bell polynomials. The results provide a unified framework for explicit computation and algorithmic verification of higher-order congruences involving reciprocal sums and related product expressions.

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Reference graph

Works this paper leans on

14 extracted references · 10 linked inside Pith

[14]

Zhong, S

H. Zhong, S. Chern, and T. Cai. A congruence involving the quotients of euler and its applications (iii).arXiv preprint arXiv:1604.00445, pages 1–13, 2016. 12

arXiv 2016
[1]

Akiyama and H

S. Akiyama and H. Kaneko. Curious congruences for cyclotomic polynomials.arXiv preprint arXiv:2204.11267, pages 1–10, 2022

arXiv 2022
[2]

T. Cai. A congruence involving the quotients of euler and its applications (i).Acta Arith., 103(4):313–320, 2002

2002
[3]

T. Cai, X. Fu, and X. Zhou. A congruence involving the quotients of euler and its applications (ii).Acta Arith., 130(3):203–214, 2007

2007
[4]

S. Chern. On a congruence involving harmonic series and bernoulli numbers.arXiv preprint arXiv:2110.09629, pages 1–19, 2021

arXiv 2021
[5]

R. Gy. Extended congruences for harmonic numbers.arXiv preprint arXiv:1902.05258, pages 1–32, 2019

arXiv 1902
[6]

B. Kellner. Wilson’s theorem modulo higher prime powers i: Fermat and wilson quotients.arXiv preprint arXiv:2509.05235, pages 1–13, 2025

arXiv 2025
[7]

Komatsu and G

T. Komatsu and G. Liu. Congruence properties of lehmer-euler numbers.arXiv preprint arXiv:2501.01178, pages 1–21, 2025

arXiv 2025

Show all 14 references

[8]

J. Liu. Supercongruences involving ap´ ery-like numbers and bernoulli numbers.arXiv preprint arXiv:2403.19503, pages 1–10, 2024

2024 arXiv
[9]

Ma and N

R. Ma and N. Li. Some congruences involving generalized bernoulli numbers and bernoulli poly- nomials.arXiv preprint arXiv:2211.15874, pages 6–7, 2022

2022 arXiv
[10]

Ma and N

R. Ma and N. Li. Some new curious congruences involving multiple harmonic sums.arXiv preprint arXiv:2305.07869, pages 1–12, 2023

2023 arXiv
[11]

F. Morley. Note on the congruence 2 4n ≡(−1) n(2n)!/(n!)2, where 2n+ 1 is a prime.Ann. of Math., 9(1–6):168–170, 1894
[12]

Patchkoria

I. Patchkoria. Chromatic congruences and bernoulli numbers.arXiv preprint arXiv:2406.17705, pages 1–22, 2024

2024 arXiv
[13]

Z. Sun. General congruence for bernoulli polynomials.Discrete Mathematics, 262(1–3):253–276, 2003

2003

Pith tools

Reviewed July 14, 2026 · model on record in the stance chip above.

[1] [14]

Zhong, S

H. Zhong, S. Chern, and T. Cai. A congruence involving the quotients of euler and its applications (iii).arXiv preprint arXiv:1604.00445, pages 1–13, 2016. 12

arXiv 2016

[2] [1]

Akiyama and H

S. Akiyama and H. Kaneko. Curious congruences for cyclotomic polynomials.arXiv preprint arXiv:2204.11267, pages 1–10, 2022

arXiv 2022

[3] [2]

T. Cai. A congruence involving the quotients of euler and its applications (i).Acta Arith., 103(4):313–320, 2002

2002

[4] [3]

T. Cai, X. Fu, and X. Zhou. A congruence involving the quotients of euler and its applications (ii).Acta Arith., 130(3):203–214, 2007

2007

[5] [4]

S. Chern. On a congruence involving harmonic series and bernoulli numbers.arXiv preprint arXiv:2110.09629, pages 1–19, 2021

arXiv 2021

[6] [5]

R. Gy. Extended congruences for harmonic numbers.arXiv preprint arXiv:1902.05258, pages 1–32, 2019

arXiv 1902

[7] [6]

B. Kellner. Wilson’s theorem modulo higher prime powers i: Fermat and wilson quotients.arXiv preprint arXiv:2509.05235, pages 1–13, 2025

arXiv 2025

[8] [7]

Komatsu and G

T. Komatsu and G. Liu. Congruence properties of lehmer-euler numbers.arXiv preprint arXiv:2501.01178, pages 1–21, 2025

arXiv 2025