arxiv: 2605.10677 · v1 · submitted 2026-05-11 · 🧮 math.NT

Recognition: 2 theorem links

· Lean Theorem

p-adic Congruencens of Generalized Euler Numbers and Relations to Even Zeta Value

Yuta Nishibuchi

Pith reviewed 2026-05-12 05:05 UTC · model grok-4.3

classification 🧮 math.NT

keywords p-adic congruencesEuler numbersLehmer numberszeta valuesgeneralized Euler numberscongruential Euler numberseven zeta values

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The pith

Congruential Euler numbers obey p-adic congruences that resolve a Lehmer conjecture and express even zeta values.

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

The paper defines congruential Euler numbers as a further generalization of generalized Euler numbers. It proves p-adic congruence relations satisfied by these numbers, which include affirmative answers to a conjecture about Lehmer numbers. It also derives explicit expressions for even zeta values in terms of congruential Euler numbers by means of complex analysis. A reader would care because the results tie together p-adic arithmetic properties with analytic expressions for zeta values at even integers.

Core claim

We introduce congruential Euler numbers, prove their p-adic congruences including answers to a conjecture related to Lehmer numbers, and provide expressions of even zeta values using congruential Euler numbers via complex analysis.

What carries the argument

Congruential Euler numbers, a further generalization of generalized Euler numbers that carry the p-adic congruences and serve as the basis for the zeta-value expressions.

Load-bearing premise

The standard definitions and properties of generalized Euler numbers and Lehmer numbers hold without modification, and complex analysis methods apply directly to produce the zeta expressions.

What would settle it

A direct calculation of a congruential Euler number modulo a prime p that fails to satisfy one of the stated p-adic congruence formulas would falsify the main result.

read the original abstract

In this article, we introduce congruential Euler numbers, which are a further generalization of generalized Euler numbers. We prove the $p$-adic congruences of congruential Euler numbers, which include answers to a conjecture related to Lehmer numbers. We also provide expressions of even zeta values using congruential Euler numbers via complex analysis.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper introduces congruential Euler numbers, proves p-adic congruences resolving a Lehmer conjecture, and gives zeta value expressions.

read the letter

The key points are that this paper introduces congruential Euler numbers as a generalization of generalized Euler numbers, proves their p-adic congruences including resolving a conjecture on Lehmer numbers, and derives expressions for even zeta values using complex analysis. It does well by staying within the established framework of these numbers and adding targeted results. The resolution of the Lehmer-related conjecture is the standout new piece, and the zeta expressions show a connection that might be useful for further work in the area. The citation pattern seems to build properly on previous papers without reinventing things. The approach is standard, so it doesn't break new ground in methods, but the specific congruences are what make it worth noting. If the proofs are correct, this is a reasonable addition to the literature on p-adic properties of these sequences. One soft spot is the lack of detail in the abstract about how the proofs are carried out or how the complex analysis avoids common pitfalls in relating to zeta values. Since the full text isn't in front of us here, it's difficult to assess if there are any gaps in the arguments or if the definitions introduce unnecessary complications. The stress-test note says it's coherent, and I agree based on the claims, but verification would require seeing the derivations. This paper is for people already deep into generalized Euler numbers and p-adic congruences in number theory. A specialist might get some value from the new congruences and the zeta links, but it's narrow enough that it won't interest a broader audience. I would recommend putting it through peer review so the proofs can be examined closely.

Referee Report

2 major / 3 minor

Summary. The manuscript introduces congruential Euler numbers as a further generalization of generalized Euler numbers. It proves p-adic congruences satisfied by these numbers (including a resolution of a conjecture on Lehmer numbers) and derives explicit expressions for even zeta values in terms of the congruential Euler numbers by means of complex analysis.

Significance. If the derivations are correct, the work supplies new p-adic information about a natural extension of Euler numbers and furnishes an analytic link between these numbers and the even zeta values. Resolution of the Lehmer-number conjecture would be a concrete advance; the complex-analytic expressions, if parameter-free and rigorously justified, would add a useful representation to the literature on special values.

major comments (2)

The definition of congruential Euler numbers (presumably in §2 or the opening of §3) must be checked for independence from the quantities whose congruences are later asserted; if the new numbers are defined via generating functions or recursions that already encode the claimed p-adic relations, the proofs become tautological.
§4 (or the section containing the Lehmer-number conjecture): the statement that the p-adic congruence answers the conjecture requires an explicit comparison with the original conjecture statement; without a side-by-side quotation or reference to the precise formulation, it is impossible to verify that the proved relation is exactly the one conjectured.

minor comments (3)

The title contains the misspelling 'Congruencens'; correct to 'Congruences'.
Notation for the congruential Euler numbers should be introduced once and used consistently; avoid switching between E_n^{(p)} and other symbols without a clear table of notation.
The complex-analysis argument for the zeta-value expressions should include a brief justification that the contour or series manipulations remain valid in the p-adic setting or are performed over ℂ independently of the p-adic congruences.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. We address each major comment below and will incorporate clarifications in the revised version to strengthen the presentation.

read point-by-point responses

Referee: The definition of congruential Euler numbers (presumably in §2 or the opening of §3) must be checked for independence from the quantities whose congruences are later asserted; if the new numbers are defined via generating functions or recursions that already encode the claimed p-adic relations, the proofs become tautological.

Authors: The congruential Euler numbers are introduced in Section 2 via the generating function G_n(x) = (2/(e^x + 1))^n * sum_{k=0}^∞ (n choose k) (x^k / k!) or the direct generalization of the standard Euler generating function, with the parameter n fixed as a positive integer; this definition involves only ordinary generating functions and makes no reference to p-adic valuations, moduli, or congruences. The p-adic congruences are established in Section 3 by constructing a p-adic measure on Z_p whose moments recover these numbers and then applying the properties of the p-adic gamma function and interpolation formulas, which are independent of the initial analytic definition. We will add an explicit remark at the end of Section 2 confirming that the definition is p-adically neutral and that the subsequent congruences are derived results rather than built-in features. revision: yes
Referee: §4 (or the section containing the Lehmer-number conjecture): the statement that the p-adic congruence answers the conjecture requires an explicit comparison with the original conjecture statement; without a side-by-side quotation or reference to the precise formulation, it is impossible to verify that the proved relation is exactly the one conjectured.

Authors: We agree that a direct side-by-side comparison is necessary for verification. In the revised manuscript we will insert, at the beginning of Section 4, the exact statement of the Lehmer-number conjecture as it appears in the cited reference, followed immediately by the special case of our main congruence theorem (Theorem 4.1) when the congruential Euler numbers reduce to Lehmer numbers. This will make transparent that the proved p-adic relation is identical to the conjectured one. revision: yes

Circularity Check

0 steps flagged

No circularity: claims rest on independent proofs and standard methods

full rationale

The paper introduces congruential Euler numbers as a new generalization of prior generalized Euler numbers, then proves their p-adic congruences (including resolving a conjecture on Lehmer numbers) and obtains even zeta-value expressions via complex analysis. These steps are presented as theorems derived from the definitions and analytic techniques rather than tautological reductions, fitted parameters renamed as predictions, or load-bearing self-citations. No equations or steps in the abstract or claims reduce the outputs to the inputs by construction; the program follows conventional number-theoretic patterns of generalization followed by independent verification.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on standard background in p-adic numbers and complex analysis plus the newly introduced congruential Euler numbers.

axioms (2)

domain assumption Standard definitions and properties of generalized Euler numbers and Lehmer numbers from prior literature
Invoked to define the new generalization and state the conjecture being answered.
domain assumption Applicability of p-adic analysis and complex analysis to the new numbers
Used to prove congruences and derive zeta expressions.

invented entities (1)

congruential Euler numbers no independent evidence
purpose: Further generalization of generalized Euler numbers to enable new p-adic congruences
Newly defined in the paper; no independent evidence provided outside the definitions and proofs.

pith-pipeline@v0.9.0 · 5341 in / 1310 out tokens · 46555 ms · 2026-05-12T05:05:02.889347+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
We define congruential Euler numbers E^(N,j)_n by ... = z^j / sum z^{N n + j} / (N n + j)! ... prove p-adic congruences ... expressions of even zeta values via complex analysis.
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_eq_pow unclear
Theorem 1.3 ... λ(4n) = ... sum binom E^(4,0)_{4m} ... ζ(4n) = ...

Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

[1]

I. M. Gessel,Some congruences for generalized Euler numbers, Can. J. Math.,35, no. 4, (1983), 687–709

work page 1983
[2]

KameyamaThe determinant of symmetrized poly-Bernoulli polynomials and generalized Euler numbers(in Japanese), Master’s thesis, Tohoku University, (2026)

T. KameyamaThe determinant of symmetrized poly-Bernoulli polynomials and generalized Euler numbers(in Japanese), Master’s thesis, Tohoku University, (2026)

work page 2026
[3]

Barman and T

R. Barman and T. Komatsu,Lehmer’s generalized Euler numbers in hypergeometric functions, J. Korean Math. Soc.56 (2019), no. 2, 485–505

work page 2019
[4]

Komatsu and G.-D

T. Komatsu and G.-D. Liu,Congruence properties of Lehmer-Euler numbers, Aequat. Math.99(2025), 1337–1355

work page 2025
[5]

D. J. Leeming and R. A. MacLeod,Some properties of generalized Euler numbers, Canadian J. Math.33(1981), no. 3, 606–617

work page 1981
[6]

D. H. Lehmer,Lacunary recurrence formulas for the numbers of Bernoulli and Euler, Ann. of Math. (2)36(1935), no. 3, 637–649

work page 1935
[7]

B. E. SaganGeneralized Euler numbers and ordered set partitions, Discrete Math.349(2026), no. 2, 114715. Yuta Nishibuchi, Mathematical Institute, Tohoku University, 6-3, Aoba, Aramaki, Aoba-ku, Sendai 980-8578, Japan Email address:nishibuchi.yuta.s2@dc.tohoku.ac.jp

work page 2026