分野別セミナー

Roughly speaking, a zeta function is a counting function. Well-known zeta functions in number theory include the Riemann zeta function, the zeta function attached to an algebraic variety defined over a finite field, and Selberg zeta functions. They count integral ideals of a given norm, the number of solutions over a finite field, and the equivalence classes of tailless geodesics in a compact Riemann surface, respectively. A combinatorial zeta function counts tailless geodesic cycles of a given length in a finite simplicial complex. One-dimensional complexes are graphs; attached to graphs are the well-studied Ihara zeta functions. Zeta functions attached to 2-dimensional complexes are recently obtained in joint work with Ming-Hsuan Kang, Yang Fang and Chian-Jen Wang by considering finite quotients of the Bruhat-Tits buildings associated to SL(3) and Sp(4) over a p-adic field.

The purpose of this talk is to show connections between combinatorics and number theory, using zeta functions as a theme. We shall give closed form expressions of the combinatorial zeta functions mentioned above, and compare their features with those of the zeta functions for varieties over finite fields.