Generalized Dedekind sums and transformation formulae in function fields

Generating functions and generalized Dedekind sums

We study sums of the form ∑ ζ R(ζ), where R is a rational function and the sum is over all nth roots of unity ζ (often with ζ = 1 excluded). We call these generalized Dedekind sums, since the most well-known sums of this form are Dedekind sums. We discuss three methods for evaluating such sums: The method of factorization applies if we have an explicit formula for ∏ ζ(1− xR(ζ)). Multisection ca...

Asymptotic Formulas and Generalized Dedekind Sums

Generalized Stark Formulae over Function Fields

We establish formulae of Stark type for the Stickelberger elements in the function field setting. Our result generalizes a work of Hayes and a conjecture of Gross. It is used to deduce a p-adic version of Rubin-Stark Conjecture and Burns Conjecture.

Higher Dimensional Dedekind Sums

In this paper we will study the number-theoretical properties of the expression v1 nkal rcka,, d(p; a I . . . . . an) = ( 1) n/2 ~ cot cot (1) k=l P P and of related finite trigonometric sums. In Eq. (I), p is a positive integer, a~ . . . . . a, are integers prime to p, and n is even (for n odd the sum is clearly equal to zero). There are two reasons for being interested in sums of this type. F...

Dedekind Cotangent Sums

Let a, a1, . . . , ad be positive integers, m1, . . . ,md nonnegative integers, and z1, . . . , zd complex numbers. We study expressions of the form ∑