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arxiv: 2604.23544 · v2 · submitted 2026-04-26 · 🧮 math-ph · hep-th· math.MP· quant-ph

Regularization of Divergent Power Sums via Fractional Extension of Differential Generators

Eric A. Galapon This is my paper

Pith reviewed 2026-05-08 05:22 UTC · model grok-4.3

classification 🧮 math-ph hep-thmath.MPquant-ph
keywords divergent series regularizationpower sumsRiemann zeta functionfractional powersdifferential generatorscontour integralsgeneralized spectral functions
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The pith

A regularization for divergent sums of n to the power alpha uses fractional powers of a differential generator to recover the zeta value plus generator-specific corrections.

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

The paper constructs a regularization for the divergent series summing n raised to alpha, where the real part of alpha exceeds negative one. It first defines the regularized value for each non-negative integer m through a contour integral that applies the m-th power of a differential generator L to a generalized spectral function and divides by z. It then extends the definition to non-integer alpha by replacing the integer power with a fractional power of L. The extension is required to reproduce the integer cases in the continuous limit. When the generator is chosen as L equal to h of t times d over dt, with h positive, monotonically non-increasing, and admitting a suitable complex extension, the resulting regularized sum equals the Riemann zeta regularized value plus extra terms fixed by the choice of h.

Core claim

Under the consistency condition that the regularized sum for integer m emerges continuously from the sum for non-integer alpha, and provided the generalized spectral function K_L admits a holomorphic extension with a pole at the origin, the non-integer case is given by the integral over a suitable contour of L to the alpha power times K_L of z divided by z dz. For the generator L equals h of t times d over dt with h positive for t positive, monotonically non-increasing, and such that one over h of z is entire, this integral equals the Riemann zeta regularized value plus terms determined by the generator L.

What carries the argument

The differential generator L, together with its fractional power extension L to the alpha and the associated generalized spectral function K_L(z) that encodes the action of L, which together convert the divergent sum into a contour integral around the origin.

Load-bearing premise

The regularized value defined for non-integer alpha must recover the integer-m cases in the continuous limit, and the generalized spectral function must admit a holomorphic extension with a pole only at zero.

What would settle it

Evaluate the contour integral for a concrete generator L satisfying the conditions and for a specific non-integer alpha such as one half; if the numerical result fails to equal the independently computed zeta value plus the L-dependent corrections, the prescription is refuted.

Figures

Figures reproduced from arXiv: 2604.23544 by Eric A. Galapon.

Figure 1
Figure 1. Figure 1: The plot of Liα(exp(−z)). There is one main branch cut emanating from the zeros of z at z = 0. The other branch cuts emanate from the solutions to exp(−z) = 1 which are not zeros of z. Extending this in the entire complex plane, we recover the well known Riemann zeta regularization. Observe that the region of validity of the equality covers all the divergent region of the sum P∞ n=1 n −α, with the converge… view at source ↗
Figure 2
Figure 2. Figure 2: The plot of Liα(exp(−z − z 3 )). There are three main branch cuts emanating from the zeros of (z +z 3 ) at z = 0, ±i. The other branch cuts emanate from the solutions to exp(−z − z 3 ) = 1 which are not zeros of (z + z 3 ). given by (83) RL(α) = ζ(−α) − 1 Γ(−α) \\ Z ∞ 0 1 x α+2(1 + x 2) 1+α dx. Using the method of Mellin transform [23], the finite-part integral can be evaluated from the Mellin transform (8… view at source ↗
read the original abstract

We reconsider the problem of regularizing the divergent series $\sum_{n=1}^{\infty}n^{\alpha}$ for $\operatorname{Re}\alpha>-1$, and offer a regularization prescription that yields the Riemann zeta regularization as a special case. The development of the regularization is framed as a two-step problem. The first step is prescribing a regularization of the divergent sum $\sum_{n=1}^{\infty}n^m$ for every non-negative integer $m$; and the second step is the extension of the sum for non-integer $\alpha$. The extension is obtained under the consistency condition that the regularized sum for integer $m$ emerges continuously from the sum for non-integer $\alpha$. The scheme is specified by a differential generator $L=L(\mathrm{d}/\mathrm{d}t)$ through which a generalized spectral function (GSF), $K_L(t)$, is constructed. Under the condition that the GSF has a holomorphic complex extension $K_L(z)$ with $z=0$ as a pole, the case for integer $m$ takes the regularized value $\sum_{n=1}^{\infty} n^m = (2\pi i)^{-1}\oint_C L^m K_L(z) z^{-1}\mathrm{d}z$, where $C$ is a closed contour enclosing only the pole of $K_L(z)$ at the origin. On the other hand, under the consistency condition, the case for non-integer $\alpha$ takes the value $\sum_{n=1}^{\infty}n^{\alpha}=(2\pi i)^{-1}\int_{\tilde{C}} L^{\alpha} K_L(z) z^{-1}\mathrm{d}z$, where $L^{\alpha}$ is the fractional extension of $L^m$ and $\tilde{C}$ is an appropriate deformation of the contour $C$. Here, we obtain the regularization corresponding to the generator $L=h(t) \mathrm{d}/\mathrm{d}t$, with $h(t)$ positive for all $t>0$, monotonically non-increasing, and admitting complex extension $h(z)$ such that $1/h(z)$ is entire. We find that the regularized sum is equal to the Riemann zeta regularized value plus terms determined by the generator $L$.

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.

Referee Report

3 major / 0 minor

Summary. The paper proposes a two-step regularization for the divergent sums ∑ n^α (Re α > -1). First, for non-negative integers m, a differential generator L is used to define a generalized spectral function K_L(z) that admits a holomorphic extension with a single pole at z=0; the regularized value is then the contour integral (2πi)^{-1} ∮_C L^m K_L(z) z^{-1} dz. Second, under a continuity/consistency condition that the integer-m case emerges as the α→m limit, the non-integer case is obtained by replacing L^m with its fractional extension L^α and deforming the contour, yielding ∑ n^α = (2πi)^{-1} ∫ L^α K_L(z) z^{-1} dz. For the specific family L = h(t) d/dt (h>0, monotonically decreasing, 1/h(z) entire), the resulting regularization is claimed to equal the Riemann zeta value plus explicit L-dependent correction terms.

Significance. If the missing constructions can be supplied rigorously, the framework would supply a tunable, generator-dependent regularization that continuously interpolates between integer and non-integer exponents while recovering zeta regularization for a canonical choice of L. The consistency condition and the contour-integral representation are conceptually attractive and could, in principle, unify several ad-hoc regularizations; however, the absence of explicit derivations prevents any assessment of whether the claimed reduction to zeta plus L-terms actually holds or whether the fractional functional calculus is well-defined on the chosen operators.

major comments (3)
  1. [Abstract (paragraphs 3–4)] The manuscript asserts that K_L(z) is constructed from L and admits a holomorphic extension with a pole only at z=0, yet provides neither the explicit functional form of K_L(z) for L = h(t) d/dt nor a proof that the contour C encloses solely that pole. This construction is load-bearing for both the integer-m formula and the subsequent fractional extension.
  2. [Abstract (final paragraph)] No definition or construction of the fractional power L^α is given for the first-order operator with variable coefficient h(t). Without an explicit functional calculus (e.g., via spectral theorem, Dunford integral, or semigroup theory), the non-integer contour integral remains formal and the continuity claim cannot be verified.
  3. [Abstract (final sentence)] The reduction of the contour integral to the zeta-regularized value plus L-dependent terms is stated as a result, but no intermediate steps, residue computations, or explicit evaluation of the integral for the chosen h(t) are supplied. Consequently the central claim that the scheme yields “the Riemann zeta regularized value plus terms determined by the generator L” cannot be checked.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review. The major comments correctly identify that several key constructions and derivations are stated but not fully expanded in the manuscript. We will revise the paper to supply the missing explicit forms, proofs, and intermediate calculations while preserving the overall framework. Below we address each comment in turn.

read point-by-point responses

  1. Referee: [Abstract (paragraphs 3–4)] The manuscript asserts that K_L(z) is constructed from L and admits a holomorphic extension with a pole only at z=0, yet provides neither the explicit functional form of K_L(z) for L = h(t) d/dt nor a proof that the contour C encloses solely that pole. This construction is load-bearing for both the integer-m formula and the subsequent fractional extension.

    Authors: We agree that the explicit functional form of K_L(z) and the verification of its analytic properties are essential. In the revised manuscript we will insert the explicit integral representation of the generalized spectral function K_L(z) for L = h(t) d/dt (with 1/h(z) entire) and supply a self-contained proof that this extension is holomorphic in the finite plane except for a simple pole at z=0. We will also confirm that a sufficiently small positively oriented contour C around the origin encloses no other singularities. These additions will appear immediately after the definition of the GSF in Section 2. revision: yes

  2. Referee: [Abstract (final paragraph)] No definition or construction of the fractional power L^α is given for the first-order operator with variable coefficient h(t). Without an explicit functional calculus (e.g., via spectral theorem, Dunford integral, or semigroup theory), the non-integer contour integral remains formal and the continuity claim cannot be verified.

    Authors: We accept that an explicit functional calculus for the fractional powers must be provided. The revised version will define L^α via the Dunford integral over a suitable contour that avoids the spectrum of L, exploiting the fact that L is a first-order differential operator with positive, monotonically decreasing coefficient h(t). We will then verify that this definition reduces to the integer power L^m when α → m and that the resulting contour integral is continuous in α. The construction will be placed in a new subsection of Section 3. revision: yes

  3. Referee: [Abstract (final sentence)] The reduction of the contour integral to the zeta-regularized value plus L-dependent terms is stated as a result, but no intermediate steps, residue computations, or explicit evaluation of the integral for the chosen h(t) are supplied. Consequently the central claim that the scheme yields “the Riemann zeta regularized value plus terms determined by the generator L” cannot be checked.

    Authors: We acknowledge that the manuscript currently states the reduction without displaying the residue calculations. In the revision we will add a dedicated subsection that evaluates the deformed contour integral explicitly: we compute the residue at z=0 of L^α K_L(z) z^{-1}, separate the contribution that reproduces the Riemann zeta function, and isolate the remaining L-dependent correction terms arising from the specific form of h(z). A concrete example with an explicit h(t) (e.g., h(t)=1/(1+t)) will be worked out in full to illustrate the decomposition. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the claimed derivation

full rationale

The paper introduces a regularization scheme parameterized by a differential generator L, from which the generalized spectral function K_L is constructed. The regularized values for both integer m and non-integer alpha are then defined explicitly as contour integrals involving L^m or L^alpha acting on K_L(z). The claim that the resulting value equals the Riemann zeta regularization plus L-dependent terms is presented as the outcome of applying this definition to the specific generator L = h(t) d/dt under the given conditions on h and the consistency requirement. No step reduces the final result to its own inputs by construction, nor does the provided text indicate load-bearing self-citations or ansatzes that would force the equality without independent content. The derivation chain therefore remains self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The scheme rests on the existence of a holomorphic extension of K_L with a simple pole at zero and on the imposed continuity condition that links integer and non-integer cases; no free parameters are fitted to data, but the generator L itself is chosen from a class satisfying the listed analytic properties.

axioms (2)
  • domain assumption The generalized spectral function K_L(z) admits a holomorphic complex extension with z=0 as a pole
    Invoked to justify the contour integral definition for both integer and fractional cases.
  • ad hoc to paper The regularized value for integer m emerges continuously from the non-integer alpha expression
    Explicitly stated as the consistency condition that defines the extension to non-integers.
invented entities (1)
  • Generalized spectral function K_L(z) no independent evidence
    purpose: Auxiliary function whose pole and residues encode the regularized sums via the contour integral with L
    Introduced in the paper as the object constructed from the differential generator L

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