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 $L=h(t) \mathrm{d}/\mathrm{d}t$, 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 $L$.