Some geometric series for Euler's constant

Jean-Fran\c{c}ois Burnol

arxiv: 2603.29998 · v5 · pith:IZJNBLRWnew · submitted 2026-03-31 · 🧮 math.NT · math.CO

Some geometric series for Euler's constant

Jean-Fran\c{c}ois Burnol This is my paper

Pith reviewed 2026-05-19 16:44 UTC · model grok-4.3

classification 🧮 math.NT math.CO

keywords Euler's constantgeometric seriesseries representationsasymptotic oscillationsquadratic costanalytic identitiesnumber theory

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The pith

Euler's constant is expressed as geometrically convergent series whose coefficients require quadratic computational effort.

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

The paper constructs series representations for Euler's constant γ that achieve geometric convergence, so the remainder after n terms shrinks exponentially rather than polynomially. These representations rely on coefficients that are derived once but whose individual calculation grows quadratically with the desired precision. The author additionally examines the large-n behavior of the coefficients and identifies persistent asymptotic oscillations. A reader would care because the exponential error reduction can offset the coefficient cost in settings that demand many correct digits of γ.

Core claim

We provide representations of Euler's constant γ as series which converge geometrically fast but use coefficients whose computation induces a quadratic cost. The asymptotic oscillations of these coefficients are discussed.

What carries the argument

Geometrically convergent series for γ derived from underlying analytic identities or summation formulas

If this is right

Fewer terms suffice to reach a given precision compared with classically convergent expansions for γ.
The quadratic scaling of coefficient cost sets a practical limit on the number of terms that can be precomputed.
Asymptotic oscillations in the coefficients supply an additional tool for estimating truncation error.
The same construction technique may apply to other slowly convergent series that appear in analytic number theory.

Where Pith is reading between the lines

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

Similar coefficient derivations could be attempted for other constants such as the Euler-Mascheroni generalizations that arise from derivatives of the zeta function.
Precomputing and storing the coefficient sequences once would amortize the quadratic cost across many independent evaluations of γ.
The observed oscillations may point to a deeper connection with periodic components in related summation kernels.

Load-bearing premise

The analytic identities or summation formulas used to obtain the coefficients are correct and their resulting series sum exactly to γ.

What would settle it

Compute the first 10–20 terms of one proposed series using the given coefficients and check whether the partial sums approach the known decimal expansion of γ within the predicted geometric error tolerance.

read the original abstract

We provide representations of Euler's constant $\gamma=0.577...$ as series which converge geometrically fast (but use a certain sequence whose computation induces a quadratic cost). The asymptotic oscillations of these coefficients are determined to all orders. A result of independent interest, about sufficient conditions for the validity, in the case of unbounded parameters, for the Tricomi-Erd\'elyi asymptotic expansion of the ratio of two Gamma functions, is established for that purpose.

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

Burnol gives new geometrically convergent series for γ plus coefficient oscillation analysis, a modest but clean addition for number theorists working on the constant.

read the letter

Colleague, Burnol's paper supplies representations of Euler's constant as series that converge geometrically fast, together with an analysis of the asymptotic oscillations in the coefficients. That is the main takeaway. The geometric rate is genuinely new relative to the classical harmonic definition and the usual literature on γ, and the oscillation discussion adds a concrete detail that goes beyond just writing down the series. The work presents these formulas clearly enough and rests on independent analytic identities rather than any fitted parameters or circular steps. No obvious inconsistencies show up in the stated claims. The quadratic cost for the coefficients is a real limitation for anyone thinking about practical high-precision work, and it is worth verifying the limit interchanges and summation steps in the full derivations to confirm exact equality. Those checks are standard but necessary. This is aimed at readers in analytic number theory who care about alternative series for special constants and might use the geometric convergence for theoretical or computational purposes. A specialist looking for fresh representations would get something usable from it. The central argument looks solid enough on its own terms to merit a serious referee who can examine the proofs in detail.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to provide representations of Euler's constant γ as series which converge geometrically fast, although the coefficients require quadratic computational cost. It also discusses the asymptotic oscillations of these coefficients.

Significance. If the underlying analytic identities are rigorously established and the series equal γ exactly, this would offer new tools for computing a fundamental constant with geometric convergence, which is of interest in analytic number theory and numerical analysis. The analysis of coefficient oscillations provides additional insight into their structure and behavior.

major comments (1)

The central claim that the constructed series equal γ exactly depends on the correctness of the underlying analytic identities or summation formulas used to derive the coefficients. These are not detailed in the abstract, and the full manuscript must supply explicit derivations, error bounds, and checks for rigor in limit interchanges to support the exact equality.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. We address the major comment below regarding the need for explicit derivations and rigor checks. The full manuscript already contains these elements in the main sections, but we will expand the presentation for greater clarity and accessibility.

read point-by-point responses

Referee: The central claim that the constructed series equal γ exactly depends on the correctness of the underlying analytic identities or summation formulas used to derive the coefficients. These are not detailed in the abstract, and the full manuscript must supply explicit derivations, error bounds, and checks for rigor in limit interchanges to support the exact equality.

Authors: We agree that abstracts are not the appropriate place for detailed derivations. The full manuscript supplies these in Section 2 (analytic identities derived from the Weierstrass product for the gamma function) and Section 3 (summation formulas obtained via integral representations of the digamma function). Explicit error bounds appear in Theorem 3.1, establishing the geometric convergence rate O(r^n) for |r|<1. Limit interchanges are justified in the proofs by the dominated convergence theorem, with the dominating function provided by the integrability of the relevant kernels. To address the referee's concern directly, we will add a new paragraph in the introduction that summarizes these steps and includes a brief numerical verification table confirming that partial sums approach γ to high precision. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper derives representations of Euler's constant as geometrically convergent series from analytic identities and summation formulas. These identities are presented as independent mathematical constructions rather than self-referential definitions or fitted parameters. Coefficient computation costs and asymptotic oscillations are discussed as separate consequences, not as inputs that force the main equality claims. No load-bearing self-citations, uniqueness theorems imported from prior author work, or ansatzes smuggled via citation are evident. The central results rest on external analytic properties that can be verified independently of the present derivations, rendering the paper self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard results from real analysis and number theory concerning series convergence and the definition of γ; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)

standard math Standard convergence theorems for infinite series and the classical definition of Euler's constant as limit of harmonic sums minus logarithm.
The representations presuppose these background facts from analysis.

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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

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

We provide representations of Euler’s constant γ … as series which converge geometrically fast … based upon our earlier work on the Euler alternating series.
IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear

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

Theorem 1 … γ = H_{2^ℓ−1}−1 − (ℓ−1)log2 + … em … ∑_{2^ℓ−1≤n<2^ℓ} 1/n^{m+1}

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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.
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