Resolution Of Multiplicative Anomaly Of Zeta Regularization For Polynomials

Abstract

In this paper, the problem of multiplicative anomaly of zeta regularization is solved for polynomials. For a regularizable sequence $\Lambda$, we explicitly calculate the zeta regularized product of $(\Lambda-z_1)\dots(\Lambda-z_n)$ for $z_1,\dots,z_n\in\mathbb{C}$. We give an explicit formula for the discrepancies between polynomials. Our results imply Mizuno's theorem as a special case. We furthermore give novel regularized product formulas for multi-dimensional products in terms of Barnes multiple gamma functions, zeros of the Riemann zeta function and zeros of the Bessel function of the first kind.