Notes on 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

Notes on Dedekind Rings

These notes record the basic results about DVR’s (discrete valuation rings) and Dedekind rings, with at least sketches of the non-trivial proofs, none of which are hard. This is standard material that any educated mathematician with even a mild interest in number theory should know. It has often slipped through the cracks of Chicago’s first year graduate program, but then we would need at least...

Note on Dedekind Type Dc Sums

In this paper we study the Euler polynomials and functions and derive some interesting formulae related to the Euler polynomials and functions. From those formulae we consider Dedekind type DC(Daehee-Changhee)sums and prove reciprocity laws related to DC sums.

Supplementary Notes on Dedekind Cuts

Motivation: Our textbook discusses and even proves many properties of R, the field of real numbers; but it doesn’t define it. I felt that it would be rather awkward to discuss real numbers without knowing what they were and I decided to write some notes on the construction of R. The approach I am following is called ‘Dedekind cut’, discovered by a German mathematician, Richard Dedekind (1831-19...