Zeta functions of periodic cubical lattices and cyclotomic-like polynomials

Abstract

Zeta functions of periodic cubical lattices are explicitly derived by computing all the eigenvalues of the adjacency operators and their characteristic polynomials. We introduce cyclotomic-like polynomials to give factorization of the zeta function in terms of them and count the number of orbits of the Galois action associated with each cyclotomic-like polynomial to obtain its further factorization. We also give a necessary and sufficient condition for such a polynomial to be irreducible and discuss its irreducibility from this point of view.

Advanced Studies in Pure Mathematics 84, 2020, Various Aspects of Multiple Zeta Functions, in honor of Professor Kohji Matsumoto's 60th birthday, pp. 93-121