Generating Functions and Generalized Dedekind Sums

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

Dedekind Sums and Values of L-functions at Positive Integers

In this paper, we study Dedekind sums and we connect them to the mean values of Dirichlet L-functions. For this, we introduce and investigate higher order dimensional Dedekind-Rademacher sums given by the expression Sd( −→ a0 , −→ m0) = 1 a0 0 a0−1 ∑

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 ∑