Regularization of Divergent Power Sums via Fractional Extension of Differential Generators

TL;DR

This paper develops a regularization scheme for divergent power sums using fractional extensions of differential operators, unifying and generalizing existing methods like the Riemann zeta regularization.

Contribution

It introduces a novel regularization framework based on differential generators and spectral functions, extending sums for non-integer powers while ensuring consistency with integer cases.

Findings

The regularized sum for integer powers matches classical results.

The scheme generalizes Riemann zeta regularization via spectral functions.

Additional terms depend on the chosen differential generator.

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 $L$.