Regularization of Divergent Power Sums via Fractional Extension of Differential Generators

Abstract: We reconsider the problem of regularizing the divergent series $\sum_{n=1}^{\infty}n^α$ for $\operatorname{Re}α>-1$, 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 $α$. The extension is obtained under the consistency condition that the regularized sum for integer $m$ emerges continuously from the sum for non-integer $α$. 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 $\sum_{n=1}^{\infty} n^m = (2πi)^{-1}\oint_C L^m K_L(z) z^{-1}\mathrm{d}z$, 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 $α$ takes the value $\sum_{n=1}^{\infty}n^α=(2πi)^{-1}\int_{\tilde{C}} L^α K_L(z) z^{-1}\mathrm{d}z$, where $L^α$ 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 $L=h(t) \mathrm{d}/\mathrm{d}t$, where $h(t)$ has the analytic extension $h(z)$ such that $1/h(z)$ is an entire function. We find that the regularized sum is equal to the Riemann zeta regularized value plus terms determined by the generator $L$.

• The paper introduces a novel framework that employs fractional extensions of differential generators to regularize divergent power sums beyond standard zeta methods.
• It develops a generalized spectral function that recovers the Riemann zeta prescription while incorporating corrections from analytic functions such as h(t).
• By extending operators to non-integer powers, the approach ensures continuous regularization across integer and non-integer exponents, with implications in quantum field theory and spectral analysis.

Regularization of Divergent Power Sums via Fractional Extension of Differential Generators

Introduction and Motivation

The article fundamentally approaches the regularization of divergent sums of the form $\sum_{n=1}^\infty n^\alpha$ for $\operatorname{Re}\,\alpha > -1$ through the lens of generalized differential operators and their fractional powers. The main objective is to systematize a regularization prescription that encompasses the Riemann zeta regularization as a special case, while allowing a family of more general regulators parameterized by a differential generator $L$. This framework addresses physical scenarios where the traditional Riemann zeta function regularization yields physically non-intuitive results, such as the vanishing Casimir force in some fermionic systems, motivating the need for a more flexible and physically motivated regularization machinery.

Generalized Spectral Function Framework

The approach relies on associating to any regularization a differential generator $L$ acting on a set of eigenfunctions $g_n(t)$, defined by the eigenvalue equation $L g_n(t) = n g_n(t)$ with suitable boundary conditions $g_n(0)=1$ and $g_n(\infty)=0$. The generalized spectral function (GSF) is thus defined as $K_L(t) = \sum_{n=1}^\infty g_n(t)$.

For the canonical zeta prescription, $L = -\mathrm{d}/\mathrm{d}t$ yields $\operatorname{Re}\,\alpha > -1$0 and recovers the traditional spectral function $\operatorname{Re}\,\alpha > -1$1.

The regularized value of the trace identities, i.e., the divergent sum $\operatorname{Re}\,\alpha > -1$2, is assigned via an analytic prescription based on the expansion of $\operatorname{Re}\,\alpha > -1$3 about $\operatorname{Re}\,\alpha > -1$4 and extraction of the constant (finite-part) via contour integration. For arbitrary $\operatorname{Re}\,\alpha > -1$5, the Keldysh-type operator solution $\operatorname{Re}\,\alpha > -1$6 with $\operatorname{Re}\,\alpha > -1$7 analytic and non-vanishing at zero, yields $\operatorname{Re}\,\alpha > -1$8 with $\operatorname{Re}\,\alpha > -1$9. Thus, the spectral function generalizes to $L$0, transforming the kernel ($L$1) and, by extension, deforming the singularity structure in the analytic continuation.

Regularized Trace Identities and Polylogarithm Structure

The $L$2-th action of $L$3 is related via the transformation $L$4 to the shifted operator $L$5. This observation allows the direct realization

$L$6

where $L$7 is the polylogarithm, thus relating regularization to polylogarithmic structures and their analytic continuations.

The regularized value is then

$L$8

with the contour enclosing the pole at $L$9.

Employing established asymptotic expansions for the polylogarithm, this regulator separates into the Riemann zeta value (i.e., $L$0) plus a term encoding the choice of $L$1—that is, the difference between this general prescription and the standard zeta regularization is entirely captured by derivatives of $L$2 at $L$3. Explicit examples for $L$4, etc. are given in terms of $L$5.

Fractional Extension of Generators: Non-Integer Powers

A central technical contribution of the work is formulating a fractional calculus extension $L$6 that interpolates between integer powers, in such a way that the regularized values for non-integer $L$7 are continuous in the limit $L$8.

The operatorial action $L$9 is first constructed in differential form via a generalized Stirling number expansion, and an integral representation via repeated formal inversion. Both approaches agree for the spectral function acting on $g_n(t)$0, yielding

$g_n(t)$1

and, for the spectral kernel,

$g_n(t)$2

valid for all complex $g_n(t)$3.

The analyticity properties of $g_n(t)$4 and the deformation of contour integrals to respect branch cuts and singularities are meticulously discussed. In particular, for $g_n(t)$5, the singularity at $g_n(t)$6 remains a branch point, necessitating Hankel-type integral prescriptions.

Figure 1: The plot of $g_n(t)$7; the main branch cut at $g_n(t)$8 and secondary cuts from $g_n(t)$9 highlight the analytic structure controlling the contour deformation for regularization.

Hankel-Type Regulators and Explicit Integral Representations

For suitable $L g_n(t) = n g_n(t)$0, the analytic structure allows a deformation to a Hankel contour along the negative real axis. The prescription reads

$L g_n(t) = n g_n(t)$1

recovering the standard zeta value when $L g_n(t) = n g_n(t)$2 (i.e., $L g_n(t) = n g_n(t)$3).

For more involved $L g_n(t) = n g_n(t)$4, the additional term is nonvanishing, resulting in a scheme that is explicitly dependent on the generator and able to yield physically distinct results. The same framework also allows the use of (finite-part) regularized Mellin transforms to express the correction term.

An explicit worked example is given for $L g_n(t) = n g_n(t)$5, yielding polynomial $L g_n(t) = n g_n(t)$6 and nontrivial multi-branch-point structure.

Figure 2: The plot of $L g_n(t) = n g_n(t)$7, illustrating additional branch cuts from zeros of $L g_n(t) = n g_n(t)$8 and enhanced analytic complexity for the regulator.

Recovery of the Riemann Zeta Regularization

For the canonical case $L g_n(t) = n g_n(t)$9, all correction terms vanish, i.e., $g_n(0)=1$0 and all derived objects revert to the Riemann zeta prescription. Thus, $g_n(0)=1$1 over the required domain.

A noteworthy byproduct is a new Hankel-type integral representation of the Riemann zeta function in terms of polylogarithm transforms,

$g_n(0)=1$2

valid for $g_n(0)=1$3.

Finite-Part versus Regularized Limit

The paper rigorously distinguishes between the finite-part regularization (term-by-term subtraction of divergences in small-$g_n(0)=1$4 asymptotics) and the regularized limit as implemented via the contour prescription. It is shown that mere finite-part extraction breaks continuity between non-integer and integer $g_n(0)=1$5, thus failing the physically motivated requirement that the regulator behaves continuously in the $g_n(0)=1$6 limit. The contour regularization prescription ensures this consistency, at the cost of more intricate corrections for integer indices.

Implications and Outlook

The results presented establish an infinite family of regularizations for divergent series, parameterized by the choice of differential generator $g_n(0)=1$7, all consistent with analytic continuation and fractional calculus. This scheme allows adaptation to specific physical requirements, e.g., the sum $g_n(0)=1$8 resulting in nonzero values in physically motivated alternatives to the standard zeta regularization.

Importantly, the method generalizes to power sums over arbitrary monotonic spectra $g_n(0)=1$9 by choosing $g_n(\infty)=0$0 such that $g_n(\infty)=0$1 is well-posed, providing a systematic path to a gamut of regularizations beyond the physical and mathematical domains already addressed in the spectral zeta formalism.

The article also addresses the analytic caveats: for certain spectral sequences, e.g., $g_n(\infty)=0$2, the GSF may possess natural boundaries that thwart analytic continuation; such situations will require further technical refinement.

From a functional analysis perspective, full rigor can be obtained via sectorial operators on Banach spaces, where the Balakrishnan calculus is applicable. The isospectrality with the reference first-order generator allows for reinterpreting regularization in operator-theoretic terms.

Conclusion

The fractional extension of differential generators yields a unifying framework for regularizing divergent power sums, subsuming and generalizing the Riemann zeta function prescription. This formalism not only provides analytic control and flexibility but is amenable to explicit calculation via polylogarithm and Mellin transform analytical machinery. Its implications are immediate in fields requiring regularization of divergent series—quantum field theory (spectral determinants, Casimir energy), mathematical physics (trace identities), and analytic number theory—while also hinting at deeper relations between operator theory, spectral analysis, and regularization procedures.

The principal open issue is the selection of the “physically correct” regulator $g_n(\infty)=0$3 in any application. The work suggests that physical constraints—possibly derived from dynamical, boundary, or symmetry conditions—will guide this choice, ultimately linking the abstract theory with empirical observables.