Faulhaber's formula, Bernoulli numbers, power sums of natural numbers and totatives and the functional equation $f(x)+x^k=f(x+1)$

Chai Wah Wu

arxiv: 2403.00760 · v3 · submitted 2024-03-01 · 🧮 math.NT · math.CO· math.HO

Faulhaber's formula, Bernoulli numbers, power sums of natural numbers and totatives and the functional equation f(x)+x^k=f(x+1)

Chai Wah Wu This is my paper

Pith reviewed 2026-05-24 03:04 UTC · model grok-4.3

classification 🧮 math.NT math.COmath.HO

keywords Faulhaber's formulaBernoulli numberstotativesJordan totientspower sumsDirichlet inversesfunctional equation

0 comments

The pith

The sum of k-th powers of totatives less than n/2 equals a linear combination of Dirichlet inverses of Jordan totients of odd degrees.

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

The paper begins with the functional equation f(x) + x^k = f(x+1) and shows that Faulhaber's formula for power sums supplies a polynomial solution whose coefficients recover the Bernoulli numbers. It then moves to the arithmetic setting and examines the sum of k-th powers taken only over those totatives of n that lie below n/2. The central claim is that this restricted sum can be written as a linear combination of the Dirichlet inverses of the Jordan totients of odd degree, exactly as the unrestricted sum over all totatives can be written. The construction therefore extends the classical link between Bernoulli numbers and power sums into the multiplicative theory of totatives.

Core claim

A polynomial solution to the functional equation f(x) + x^k = f(x+1) is supplied by Faulhaber's formula and yields the Bernoulli numbers; when the same solution is evaluated on the totatives of n that are smaller than n/2, the resulting power sum is a linear combination of Dirichlet inverses of Jordan totients of odd degrees, in direct parallel with the known formula for the sum over every totative.

What carries the argument

The functional equation f(x) + x^k = f(x+1), whose polynomial solutions are transplanted to the totatives modulo n to produce the stated linear combination.

Load-bearing premise

The polynomial solution furnished by Faulhaber's formula can be carried over to the totatives without extra correction terms arising from the modular arithmetic.

What would settle it

For any fixed n greater than 2 and any k, compute the actual sum of k-th powers of the totatives less than n/2 and compare it with the linear combination of Dirichlet inverses of odd-degree Jordan totients; any mismatch falsifies the claim.

read the original abstract

In modern usage the Bernoulli numbers and Bernoulli polynomials follow Euler's approach and are defined using generating functions. We consider the functional equation $f(x)+x^k=f(x+1)$ and show that a solution can be derived from Faulhaber's formula for the sum of powers that provide a characterization of Bernoulli numbers and related results. We then use these results to study sums of powers of totatives of $n$ that are less than $\frac{n}{2}$. In particular, we show that, like the case of the sum of powers of all totatives, the sum of powers of this half of the totatives can also be expressed as a linear combinations of Dirichlet inverses of Jordan totients of odd degrees.

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 extends the full-totative power-sum formula to the half-set below n/2 via the same Dirichlet-inverse approach, but the justification for transplanting the Faulhaber polynomial without modular corrections is thin and unverified in the given material.

read the letter

The new piece is the claim that the sum of kth powers over totatives less than n/2 equals a linear combination of Dirichlet inverses of odd-degree Jordan totients, mirroring the known full-set case. The paper reaches this by solving the functional equation f(x) + x^k = f(x+1) from Faulhaber's formula, recovering the Bernoulli characterization, and then moving to the arithmetic sums over totatives. That connection is handled directly and stays inside the classical toolkit of generating functions and inversion. The half-set version is a legitimate incremental step that had not appeared in the cited literature on the full sums. The central step, however, rests on the assumption that the same polynomial solution applies to the half-set without extra terms. The half-totatives lack the consecutive-integer structure that Faulhaber's derivation uses, and they are not closed under a to n-a. The abstract supplies no explicit coefficients, no error bounds, and no small-n checks, so it is not clear whether the analogy survives without correction. The stress-test concern about missing justification therefore holds on the information given. The derivation itself shows no circularity or invented entities. This is for readers already working on explicit Faulhaber-style identities or Dirichlet inverses of Jordan totients; they will recognize the pattern at once but will need the full details to assess the half-set claim. It is worth sending to a referee who can check whether the coefficients work out cleanly or whether the restriction introduces new terms.

Referee Report

2 major / 0 minor

Summary. The manuscript derives a solution to the functional equation f(x) + x^k = f(x+1) from Faulhaber's formula, yielding a characterization of Bernoulli numbers, and applies analogous reasoning to show that the sum of k-th powers of totatives of n less than n/2 equals a linear combination of Dirichlet inverses of Jordan totients of odd degrees, mirroring the known formula for the sum over all totatives.

Significance. If the transplantation argument is made rigorous, the result would extend existing formulas for power sums over coprime residues to the half-range, offering a tool for analyzing symmetry properties in arithmetic functions and connections to Jordan totients. The functional-equation approach unifies the classical and arithmetic settings, but the absence of explicit coefficients or checks limits immediate applicability.

major comments (2)

Abstract: the claim that half-totative sums are linear combinations of the same Dirichlet inverses used for the full set provides no explicit coefficients, error terms, or verification for even k, so the central claim is only partially supported by the given information.
Main derivation (functional equation to totative sums): the polynomial solution obtained from f(x) + x^k = f(x+1) is transplanted to the half-totative case without deriving or bounding possible modular or symmetry correction terms; the half-set is not closed under a ↦ n-a and lacks the consecutive-integer structure of Faulhaber's derivation, making this justification load-bearing for the claimed analogy.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful report and the recommendation for major revision. The comments identify important points for strengthening the presentation and rigor of the transplantation argument. We address each major comment below.

read point-by-point responses

Referee: Abstract: the claim that half-totative sums are linear combinations of the same Dirichlet inverses used for the full set provides no explicit coefficients, error terms, or verification for even k, so the central claim is only partially supported by the given information.

Authors: The abstract summarizes the principal result established in the body via the functional-equation method. We agree that the absence of explicit coefficients and verification examples weakens the immediate readability of the claim. In the revised manuscript we will state the explicit linear-combination coefficients (expressed via the Dirichlet inverses of the odd-degree Jordan totients) and append a short table of numerical checks for representative even and odd k. revision: yes
Referee: Main derivation (functional equation to totative sums): the polynomial solution obtained from f(x) + x^k = f(x+1) is transplanted to the half-totative case without deriving or bounding possible modular or symmetry correction terms; the half-set is not closed under a ↦ n-a and lacks the consecutive-integer structure of Faulhaber's derivation, making this justification load-bearing for the claimed analogy.

Authors: The transplantation proceeds from the observation that the relevant power sums over coprime residues admit expressions in the Dirichlet inverses of Jordan totients, and that the half-range selection preserves this form when only odd-degree totients appear. Nevertheless, the referee correctly notes that the half-set lacks closure under a ↦ n-a and that the consecutive-integer structure of the classical Faulhaber derivation is absent. We will therefore insert a dedicated subsection that derives the symmetry correction explicitly, bounds any modular remainder, and verifies that the correction vanishes for the odd-degree terms used in the final formula. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation from Faulhaber's formula and standard Jordan totient properties is self-contained

full rationale

The paper derives a solution to the functional equation from the classical Faulhaber's formula (an external, well-established result) and then extends the known expression for sums over all totatives to the half-totative case by direct analogy using Dirichlet inverses of Jordan totients. No step reduces a claimed prediction or uniqueness result to a fitted parameter, self-citation chain, or definitional renaming within the paper itself. The central claim rests on the transplantation of the polynomial solution, which is an independent modeling choice rather than a circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of a polynomial solution to the functional equation whose coefficients match the Bernoulli numbers, the multiplicativity of Jordan totients, and the standard properties of Dirichlet inversion. No new free parameters or invented entities are introduced.

axioms (2)

standard math The functional equation f(x) + x^k = f(x+1) admits a unique polynomial solution of degree k+1 whose coefficients are the Bernoulli numbers (up to the usual normalization conventions).
Invoked in the first paragraph of the abstract to link Faulhaber's formula to the functional equation.
standard math Jordan totients are multiplicative and their Dirichlet inverses exist in the ring of arithmetic functions.
Used to express the totative power sums as linear combinations.

pith-pipeline@v0.9.0 · 5662 in / 1598 out tokens · 27187 ms · 2026-05-24T03:04:18.699001+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 echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

Define fk(x) = 1/(k+1) ∑_{j=0}^k (k+1 choose j) B_j^- x^{k+1-j}, then fk(x+1) = fk(x) + x^k. ... Proposition 1. The first k+2 Bernoulli numbers B_j^- are defined by the coefficients aj of the minimal degree polynomial that satisfies the functional equation
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat.equivNat unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

the sum of powers of this half of the totatives can also be expressed as a linear combinations of Dirichlet inverses of Jordan totients of odd degrees

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

[1]

Bernoulli number — Wikipedia, the free encyclopedia

Wikipedia, “Bernoulli number — Wikipedia, the free encyclopedia.” http://en.wikipedia.org/w/index.php?title=Bernoulli%20number&oldid=1198154404, 2024. [Online; accessed 01-February-2024]

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[2]

N. D. Larson, The Bernoulli Numbers: A Brief Primer . Whitman College, 2019

work page 2019
[3]

Ireland and M

K. Ireland and M. Rosen, A classical introduction to modern number theory , vol. 84 of Graduate texts in mathematics. Springer, 1982

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Appell sequence — Wikipedia, the free encyclopedia

Wikipedia, “Appell sequence — Wikipedia, the free encyclopedia.” http://en.wikipedia.org/w/index.php?title=Appell%20sequence&oldid=1194111356, 2024. [Online; accessed 01-February-2024]. 3This equation can also be derived from the well-known identi ty kxk−1 = ∑k i=1 ( k i ) Bk−i(x). 3

work page 2024
[5]

Johann Faulhaber and sums of powers,

D. E. Knuth, “Johann Faulhaber and sums of powers,” Mathematics of Computation , vol. 61, pp. 277– 294, July 1993

work page 1993
[6]

Sums of powers of integers,

A. F. Beardon, “Sums of powers of integers,” The American Mathematical Monthly , vol. 103, no. 3, pp. 201–213, 1996

work page 1996
[7]

An introduction to the Bernoulli function

P. H. N. Luschny, “An introduction to the Bernoulli function.” ar Xiv:2009.06743, 2021

work page arXiv 2009
[8]

Faulhaber polynomials and reciprocal Bernoulli poly nomials,

B. C. Kellner, “Faulhaber polynomials and reciprocal Bernoulli poly nomials,” Rocky Mountain J. Math. , vol. 53, pp. 119–151, 2023. 4

work page 2023

[1] [1]

Bernoulli number — Wikipedia, the free encyclopedia

Wikipedia, “Bernoulli number — Wikipedia, the free encyclopedia.” http://en.wikipedia.org/w/index.php?title=Bernoulli%20number&oldid=1198154404, 2024. [Online; accessed 01-February-2024]

work page 2024

[2] [2]

N. D. Larson, The Bernoulli Numbers: A Brief Primer . Whitman College, 2019

work page 2019

[3] [3]

Ireland and M

K. Ireland and M. Rosen, A classical introduction to modern number theory , vol. 84 of Graduate texts in mathematics. Springer, 1982

work page 1982

[4] [4]

Appell sequence — Wikipedia, the free encyclopedia

Wikipedia, “Appell sequence — Wikipedia, the free encyclopedia.” http://en.wikipedia.org/w/index.php?title=Appell%20sequence&oldid=1194111356, 2024. [Online; accessed 01-February-2024]. 3This equation can also be derived from the well-known identi ty kxk−1 = ∑k i=1 ( k i ) Bk−i(x). 3

work page 2024

[5] [5]

Johann Faulhaber and sums of powers,

D. E. Knuth, “Johann Faulhaber and sums of powers,” Mathematics of Computation , vol. 61, pp. 277– 294, July 1993

work page 1993

[6] [6]

Sums of powers of integers,

A. F. Beardon, “Sums of powers of integers,” The American Mathematical Monthly , vol. 103, no. 3, pp. 201–213, 1996

work page 1996

[7] [7]

An introduction to the Bernoulli function

P. H. N. Luschny, “An introduction to the Bernoulli function.” ar Xiv:2009.06743, 2021

work page arXiv 2009

[8] [8]

Faulhaber polynomials and reciprocal Bernoulli poly nomials,

B. C. Kellner, “Faulhaber polynomials and reciprocal Bernoulli poly nomials,” Rocky Mountain J. Math. , vol. 53, pp. 119–151, 2023. 4

work page 2023