Summary of Siegel pole results

In number theory there is a known zero-free region for a Dedekind zeta function, which can be given explicitly except for the possibility of a single first-order real zero within this region. This possible exceptional zero has come to be known as a Siegel zero and is closely connected with the Brauer–Siegel theorem on the growth of the class number times the regulator with the discriminant. See Lang [73] for more information on the implications of the non-existence of Siegel zeros. There is no known example of a Siegel zero for Dedekind zeta functions. In number fields, a Siegel zero (should it exist) “deserves” to arise already in a quadratic extension of the base field. This has now been proved in many cases (see Stark [116]).

The reciprocal of the Ihara zeta function, ζX(u)−1, is a polynomial with a finite number of zeros. Thus there is an ∈ > 0 such that any pole of ζX(u) in the region RX ≤ ∣u∣ < RX + ∈ must lie on the circle ∣u∣ = RX. This gives us the graph theoretic analog of a pole-free region, ∣u∣ < RX + ∈; the only exceptions lie on the circle ∣u∣ = RX. We will show that ζX(u) is a function of uδ with δ = δX a positive integer from Definition 23.2 below.