VALUES OF ZETA FUNCTIONS OF ARITHMETIC SURFACES AT

Algebraic Independence of Arithmetic Gamma Values and Carlitz Zeta Values

We consider the values at proper fractions of the arithmetic gamma function and the values at positive integers of the zeta function for Fq [θ] and provide complete algebraic independence results for them.

Rational values of Weierstrass zeta functions

We answer a question of Masser by showing that for the Weierstrass zeta function ζ corresponding to a given lattice Λ, the density of algebraic points of absolute multiplicative height bounded by T and degree bounded by k lying on the graph of ζ, restricted to an appropriate domain, does not exceed c(log T ), for an effective constant c > 0 depending on k and on Λ. Using different methods, we a...

Extreme Values of Zeta and L-functions

Computing Special Values of Partial Zeta Functions

We discuss computation of the special values of partial zeta functions associated to totally real number fields. The main tool is the Eisenstein cocycle Ψ , a group cocycle for GLn(Z); the special values are computed as periods of Ψ , and are expressed in terms of generalized Dedekind sums. We conclude with some numerical examples for cubic and quartic fields of small discriminant.

Zeta Functions for Hyperbolic Surfaces

Let M = H/Γ a “convex co-compact” hyperbolic surface consisting of a compact core and, possibly, a finite number of infinite volume funnels. We do not allowM to have cusps. (More formally, Γ is convex co-compact if (convex.hull(LΓ))/Γ is compact, where LΓ is the limit set of Γ.) We exclude the possibility that Γ is elementary (i.e. virtually cyclic). The closed geodesics on M are is one-to-one ...