Absolute zeta functions for zeta functions of quantum cellular automata

TL;DR

This paper introduces an absolute zeta function associated with quantum cellular automata, demonstrating its automorphic properties, explicit calculations for tensor-type QCA, and deriving related functional equations.

Contribution

It establishes that a new zeta function for QCA is an absolute automorphic form and connects it to multiple gamma and sine functions.

Findings

The absolute zeta function for QCA is an absolute automorphic form.

Explicit calculation of the zeta function for tensor-type QCA shows it as a multiple gamma function.

Derived the functional equation of the zeta function using multiple sine functions.

Abstract

Our previous work dealt with the zeta function for the interacting particle system (IPS) including quantum cellular automaton (QCA) as a typical model in the study of ``IPS/Zeta Correspondence". On the other hand, the absolute zeta function is a zeta function over F_1 defined by a function satisfying an absolute automorphy. This paper proves that a new zeta function given by QCA is an absolute automorphic form of weight depending on the size of the configuration space. As an example, we calculate an absolute zeta function for a tensor-type QCA, and show that it is expressed as the multiple gamma function. In addition, we obtain its functional equation by the multiple sine function.