Analysis of a Complex approximation to the Li-Keiper coefficients around the K Function

Danilo Merlini; Massimo Sala; Nicoletta Sala

arxiv: 1907.01903 · v1 · pith:4QWJ62JTnew · submitted 2019-06-30 · 🧮 math.GM

Analysis of a Complex approximation to the Li-Keiper coefficients around the K Function

Danilo Merlini , Massimo Sala , Nicoletta Sala This is my paper

Pith reviewed 2026-05-25 12:23 UTC · model grok-4.3

classification 🧮 math.GM

keywords Li-Keiper coefficientsKoebe functionperturbation methodstability conjectureclosed system of equationsEuler-Mascheroni constantdiscrete derivative

0 comments

The pith

A perturbation of the Li-Keiper coefficients around the Koebe function produces a closed system of equations for them.

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

The paper introduces a perturbation for the Li-Keiper coefficients around the Koebe function and derives a closed system of equations that these coefficients must satisfy. It then examines some solutions of this system that relate to discrete derivatives and presents numerical evidence supporting a conjecture that the small fluctuating component of the coefficients remains bounded by the product of the Euler-Mascheroni constant and n. A reader would care because these coefficients encode information about the zeros of the Riemann zeta function, so a closed description could simplify analysis of their growth and stability.

Core claim

By constructing a perturbation around the Koebe function, the authors obtain a closed system of equations for the Li-Keiper coefficients. Numerical solutions of this system are checked for consistency with the discrete derivative of order n, and computations support the conjecture that the tiny part lambda-tiny(n) is bounded in absolute value by gamma times n.

What carries the argument

The perturbation around the Koebe function that yields the closed system of equations for the Li-Keiper coefficients.

If this is right

The system admits multiple solutions, some of which correspond to the discrete derivative of order n of a function.
Selected solutions can be verified for correctness against known properties.
The stability conjecture implies that fluctuations lambda-tiny(n) do not grow faster than gamma n.
Numerical findings are consistent with the bound holding for the examined cases.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The method might extend to perturbations around other reference functions to test similar closed systems.
Confirmation of the bound could motivate seeking an analytic proof of the stability conjecture.
If the closed system is exact, it may allow recursive computation of higher coefficients without direct zeta function evaluations.

Load-bearing premise

The perturbation constructed around the Koebe function produces a closed system whose solutions correctly capture the behavior of the actual Li-Keiper coefficients.

What would settle it

Direct computation of lambda-tiny(n) for successively larger n and observation of whether its absolute value ever exceeds gamma times n.

read the original abstract

We introduce a kind of "perturbation" for the Li-Keiper coefficients around the Koebe function (the K function) and establish a closed system of Equations for the Li-Keiper coefficients. We then check the correctness of some of the many possible solutions offered by the system ,related to the discrete derivative of order n of a function. We also report numerical finding which support our stability conjecture that the tiny part lambda-tiny(n) (the fluctuations around the trend) are bounded in absolute values by gammaxn, where gamma is the Euler-Mascheroni constant.

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 sets up a closed system via Koebe perturbation for Li-Keiper coefficients but never shows the solutions match the standard definition from the xi function.

read the letter

The main takeaway is that the authors perturb around the Koebe function to produce a closed system of equations for the Li-Keiper coefficients and then run numerical checks on solutions of that system to support a stability bound on the fluctuations. They also look at some solutions tied to discrete derivatives of a function. That is the extent of what is new here. The numerical findings back the claim that the tiny part stays bounded by gamma times n inside their model. The paper does a reasonable job of laying out one possible self-contained system and checking a few cases numerically. The bound itself looks plausible from the reported runs. The central soft spot is exactly the one in the stress-test note. Nothing derives or checks that the model solutions equal the actual coefficients defined by the usual formula with the logarithmic derivative of xi at s=1. All the numerics stay inside the model, so they do not confirm agreement with independently computed real values. There is also no error analysis or description of the numerical method, which leaves the support hard to evaluate. This work would mainly interest people already looking at univalent-function connections to analytic number theory or at alternative representations of the Li coefficients. A reader hunting for new computational tools or verified bounds on the genuine coefficients will not find usable results. I would not bring the paper to a reading group. The missing equivalence makes the stability claim irrelevant to the actual objects. I would not cite it. The paper does not deserve peer review in its current form because the core construction needs that link established first.

Referee Report

2 major / 2 minor

Summary. The paper introduces a perturbation of the Li-Keiper coefficients around the Koebe function to derive a closed system of equations for these coefficients. It examines particular solutions of this system that correspond to discrete derivatives of order n, and presents numerical results supporting a stability conjecture that the fluctuation component λ_tiny(n) satisfies |λ_tiny(n)| ≤ γ n, with γ the Euler-Mascheroni constant.

Significance. If the derived closed system were shown to be satisfied by the standard Li-Keiper coefficients defined from the logarithmic derivative of the Riemann xi function, the construction could supply an alternative analytic framework for studying the coefficients and their connection to the Riemann hypothesis. The numerical support for the bounded-fluctuation conjecture would then become relevant to the actual coefficients rather than only to the model system.

major comments (2)

[Closed system derivation] The manuscript asserts that the perturbation around the Koebe function produces a closed system whose solutions are the Li-Keiper coefficients λ_n, yet no derivation is supplied showing that these equations are satisfied by the standard definition λ_n = 1/(n-1)! [d^{n-1}/ds^{n-1} (s^{n-1} log ξ(s))]_{s=1}. This equivalence is load-bearing for every subsequent claim about the coefficients.
[Numerical results] Numerical checks and the stability conjecture are performed exclusively on solutions of the model system; no comparison is made to independently computed values of the actual Li-Keiper coefficients obtained from the xi-function definition. Consequently the reported numerical findings do not test the conjecture for the coefficients the paper claims to study.

minor comments (2)

[Abstract] The abstract contains minor grammatical issues (e.g., “a kind of perturbation”, “numerical finding”, “gammaxn”) that should be corrected for clarity.
Notation for the fluctuation component (“lambda-tiny(n)”) and the bound (“gammaxn”) is introduced without an explicit definition or reference to the section where it is defined.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed report and the identification of two key points that require clarification. We address each major comment below and will revise the manuscript to improve precision regarding the scope of the closed system and the numerical evidence.

read point-by-point responses

Referee: [Closed system derivation] The manuscript asserts that the perturbation around the Koebe function produces a closed system whose solutions are the Li-Keiper coefficients λ_n, yet no derivation is supplied showing that these equations are satisfied by the standard definition λ_n = 1/(n-1)! [d^{n-1}/ds^{n-1} (s^{n-1} log ξ(s))]_{s=1}. This equivalence is load-bearing for every subsequent claim about the coefficients.

Authors: The closed system is obtained by introducing a perturbation of the coefficients around the Koebe function and imposing consistency conditions that close the equations. The manuscript presents this system as governing the Li-Keiper coefficients in the perturbed setting. We agree that an explicit verification that the standard definition from the xi-function logarithmic derivative satisfies the resulting equations is not supplied. In the revision we will add a dedicated subsection that either derives the equivalence from the perturbation ansatz or states the precise conditions under which the standard coefficients are expected to obey the system, thereby making the logical status of the claim transparent. revision: yes
Referee: [Numerical results] Numerical checks and the stability conjecture are performed exclusively on solutions of the model system; no comparison is made to independently computed values of the actual Li-Keiper coefficients obtained from the xi-function definition. Consequently the reported numerical findings do not test the conjecture for the coefficients the paper claims to study.

Authors: The numerical experiments and the reported bound |λ_tiny(n)| ≤ γ n are performed on particular solutions of the closed model system (those corresponding to discrete derivatives of order n). We concur that these results therefore test the stability conjecture only inside the model and do not yet constitute evidence for the actual Li-Keiper coefficients. The revision will explicitly distinguish the model conjecture from any claim about the xi-function coefficients and, where space permits, will include a short comparison with tabulated values of the standard λ_n to illustrate the qualitative similarity or difference. revision: partial

Circularity Check

0 steps flagged

No circularity detected; no load-bearing steps reduce to inputs by construction

full rationale

The abstract introduces a perturbation around the Koebe function to form a closed system asserted to govern Li-Keiper coefficients, followed by numerical checks on solutions and a stability conjecture. No equations, definitions, or derivations are supplied in the available text that would allow exhibition of any reduction (self-definitional, fitted-input, or self-citation). Without quoted paper content showing a parameter fit renamed as prediction or an ansatz imported via self-citation, the derivation cannot be shown to be equivalent to its inputs. The work is therefore treated as self-contained for the purpose of this analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5624 in / 1093 out tokens · 30259 ms · 2026-05-25T12:23:56.104443+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/Foundation/ArithmeticFromLogic.lean LogicNat.induction; embed_add; toNat_add echoes

?

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

λ(n) = ∑ (-1)^{k-n+1} binom(n,k) λ(k) + Δ (Eq. 12); λ_n = (-1)^n ∑ (-1)^k binom(2n,n-k) λ_k (Eq. 18); discrete derivative Δ^n f(m) = ∑ (-1)^k binom(n,k) f(m+(n-k)) (Appendix 3)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel; dAlembert_cosh_solution_aczel unclear

?

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

perturbation around Koebe function K(z)=z/(1-z)^2 yielding an = λ_tiny(n)/(n γ) < 1; recurrence λ_tiny(n) = 2 λ_tiny(n-1) - λ_tiny(n-2) (Eq. 9, Approximation A)

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

12 extracted references · 12 canonical work pages · 3 internal anchors

[1]

Effective method of computing Li's coefficients and their properties

K. Maslanka; “Effective method of computing Li's coefficients and their properties”, ArX3iv:math/0402168v5 (math NT)

work page arXiv
[2]

Yet Another Representation for Reciprocals of the Nontrivial Zeros of the Riemann Zeta Function

Yu.V . Matiyasevich: “Yet Another Representation for Reciprocals of the Nontrivial Zeros of the Riemann Zeta Function”,Mat.Zametki,97:3(2015), 476-479

work page 2015
[3]

Power Series Expansions of Riemann's function

J.B. Keiper: “Power Series Expansions of Riemann's function”: Mathematics ,V olume 58 ,number 198,April 1992,765-773

work page 1992
[4]

Rigorous high-precision computation of the Hurwitz zeta function and its derivatives

F. Johansson: “Rigorous high-precision computation of the Hurwitz zeta function and its derivatives”, ArXiv:1309.2877v1 (cs.SC) (2013)

work page internal anchor Pith review Pith/arXiv arXiv 2013
[5]

Fluctuation around the Gamma function and a Conjecture

D. Merlini, M. Sala, and N. Sala: “Fluctuation around the Gamma function and a Conjecture”, IOSR International Journal of Mathematics, 2019, volume 15 issue I,57-70 . 6 .V . E. Tarasov: “Exact Discrete Analogs of Derivatives of Integer Orders: Differences as Infinite Series”, Journal of Mathematics, V ol 2015 Article ID 134842

work page 2019
[6]

On the de Branges theorem

W. Koepf and D.Schmersau (1996) “On the de Branges theorem”, Complex Variables, Theory and Application: An International Journal, 31:3, 213-230

work page 1996
[7]

A numerical comment on the tiny oscillations and a heuristic conjecture

D. Merlini, M. Sala, and N. Sala: “A numerical comment on the tiny oscillations and a heuristic conjecture”, ArXiv; 1903.623v2 (math NT). (2019)

work page 1903
[8]

Quasi Fibonacci approximation to the low tiny fluctuations of the Li-Keiper coefficients: a numerical computation

D. Merlini, M. Sala, and N. Sala: “Quasi Fibonacci approximation to the low tiny fluctuations of the Li-Keiper coefficients; a numerical computation”, ArXiv :1904.07005 (math GM)(2019)

work page internal anchor Pith review Pith/arXiv arXiv 1904
[9]

A sharpening of Li's criterion for the Riemann Hypothesis

A.V oros: “Sharpening of Li's criterion for the Riemann Hypothesis”, ArXiv:math/040421342 (math. N.T.), 2004

work page internal anchor Pith review Pith/arXiv arXiv 2004
[10]

Retrieved, 10 May 2019: http://arblib.org/examples.html Appendix 1 The K function i.e

Example programs-ARB2.17-gitdocumentation, Li-Coefficients. Retrieved, 10 May 2019: http://arblib.org/examples.html Appendix 1 The K function i.e. the Koebe function was important in the de Branges proof of the Bieberbach conjecture (de Branges’s Theorem). In the variable s it is given by K(s) = s.(s-1); thus in the variable z=1-1/s, that is s=1/(1-z) we ...

work page 2019
[11]

Below we present the plots of Re(f(s) and Im(f(s) that is real and imaginary part of f(s) on the straight line defined above

We show that on the straight line s= b+i, bϵ[1,∞[ ,the function f of Eq.(1), is univalent. Below we present the plots of Re(f(s) and Im(f(s) that is real and imaginary part of f(s) on the straight line defined above. Fig. A.1 In red Re(f( b+i)) where b=Re(s) =Re(b+i.t),which is not injective and in green Im(f(s)) = Im(f(b+i)) which is injective. Thus f(s)...

work page
[12]

b= constant s = 1+i +i.t

A second case is that of a straight line, i.e. b= constant s = 1+i +i.t. Fig.A.2 In red Re(f(s)), in green Im(f(s)) in the interval t= 0-4.5 Im(f(s)) is not injective ; Re(f(s)) is not injective in 0-9.8 but “injective in 0-4.5. The function is univalent in 0-4.5. The same in the interval 4.5-∞ where Im(f(s)) is injective. Fig. A.3 In red Re(f(s)), in gre...

work page

[1] [1]

Effective method of computing Li's coefficients and their properties

K. Maslanka; “Effective method of computing Li's coefficients and their properties”, ArX3iv:math/0402168v5 (math NT)

work page arXiv

[2] [2]

Yet Another Representation for Reciprocals of the Nontrivial Zeros of the Riemann Zeta Function

Yu.V . Matiyasevich: “Yet Another Representation for Reciprocals of the Nontrivial Zeros of the Riemann Zeta Function”,Mat.Zametki,97:3(2015), 476-479

work page 2015

[3] [3]

Power Series Expansions of Riemann's function

J.B. Keiper: “Power Series Expansions of Riemann's function”: Mathematics ,V olume 58 ,number 198,April 1992,765-773

work page 1992

[4] [4]

Rigorous high-precision computation of the Hurwitz zeta function and its derivatives

F. Johansson: “Rigorous high-precision computation of the Hurwitz zeta function and its derivatives”, ArXiv:1309.2877v1 (cs.SC) (2013)

work page internal anchor Pith review Pith/arXiv arXiv 2013

[5] [5]

Fluctuation around the Gamma function and a Conjecture

D. Merlini, M. Sala, and N. Sala: “Fluctuation around the Gamma function and a Conjecture”, IOSR International Journal of Mathematics, 2019, volume 15 issue I,57-70 . 6 .V . E. Tarasov: “Exact Discrete Analogs of Derivatives of Integer Orders: Differences as Infinite Series”, Journal of Mathematics, V ol 2015 Article ID 134842

work page 2019

[6] [6]

On the de Branges theorem

W. Koepf and D.Schmersau (1996) “On the de Branges theorem”, Complex Variables, Theory and Application: An International Journal, 31:3, 213-230

work page 1996

[7] [7]

A numerical comment on the tiny oscillations and a heuristic conjecture

D. Merlini, M. Sala, and N. Sala: “A numerical comment on the tiny oscillations and a heuristic conjecture”, ArXiv; 1903.623v2 (math NT). (2019)

work page 1903

[8] [8]

Quasi Fibonacci approximation to the low tiny fluctuations of the Li-Keiper coefficients: a numerical computation

D. Merlini, M. Sala, and N. Sala: “Quasi Fibonacci approximation to the low tiny fluctuations of the Li-Keiper coefficients; a numerical computation”, ArXiv :1904.07005 (math GM)(2019)

work page internal anchor Pith review Pith/arXiv arXiv 1904

[9] [9]

A sharpening of Li's criterion for the Riemann Hypothesis

A.V oros: “Sharpening of Li's criterion for the Riemann Hypothesis”, ArXiv:math/040421342 (math. N.T.), 2004

work page internal anchor Pith review Pith/arXiv arXiv 2004

[10] [10]

Retrieved, 10 May 2019: http://arblib.org/examples.html Appendix 1 The K function i.e

Example programs-ARB2.17-gitdocumentation, Li-Coefficients. Retrieved, 10 May 2019: http://arblib.org/examples.html Appendix 1 The K function i.e. the Koebe function was important in the de Branges proof of the Bieberbach conjecture (de Branges’s Theorem). In the variable s it is given by K(s) = s.(s-1); thus in the variable z=1-1/s, that is s=1/(1-z) we ...

work page 2019

[11] [11]

Below we present the plots of Re(f(s) and Im(f(s) that is real and imaginary part of f(s) on the straight line defined above

We show that on the straight line s= b+i, bϵ[1,∞[ ,the function f of Eq.(1), is univalent. Below we present the plots of Re(f(s) and Im(f(s) that is real and imaginary part of f(s) on the straight line defined above. Fig. A.1 In red Re(f( b+i)) where b=Re(s) =Re(b+i.t),which is not injective and in green Im(f(s)) = Im(f(b+i)) which is injective. Thus f(s)...

work page

[12] [12]

b= constant s = 1+i +i.t

A second case is that of a straight line, i.e. b= constant s = 1+i +i.t. Fig.A.2 In red Re(f(s)), in green Im(f(s)) in the interval t= 0-4.5 Im(f(s)) is not injective ; Re(f(s)) is not injective in 0-9.8 but “injective in 0-4.5. The function is univalent in 0-4.5. The same in the interval 4.5-∞ where Im(f(s)) is injective. Fig. A.3 In red Re(f(s)), in gre...

work page