[Paper Review] An analogue of the Rademacher function for generalized Dedekind sums in higher dimension

Hi-joon Chae, Byungheup Jun+1 more|arXiv (Cornell University)|Jun 13, 2014

Advanced Mathematical Identities36 references3 citations

TL;DR

This paper introduces a higher-dimensional analogue of the Rademacher function for generalized Dedekind sums in $n$ dimensions, using Todd series and iterated residues to define a generalized Rademacher function that ensures integrality. It proves that the fractional parts of these generalized Dedekind sums are equidistributed via nontrivial bounds on associated exponential sums, extending classical results on Dedekind sums to higher dimensions with explicit denominator control via Bernoulli numbers.

ABSTRACT

We consider generalized Dedekind sums in dimension $n$, for fixed $n$-tuple of natural numbers, defined as sum of products of values of periodic Bernoulli functions. This includes the higher dimensional Dedekind sums of Zagier and Apostol-Carlitz' generalized Dedekind sums as well as the original Dedekind sums. These are realized as coefficients of Todd series of lattice cones and satisfy reciprocity law from the cocycle property of Todd series. Using iterated residue formula, we compute the coefficient of the decomposition of of the Todd series corresponding to a nonsingular decomposition of the lattice cone defining the Dedekind sums. We associate a Laurent polynomial which is added to generalized Dedekind sums of fixed index to make their denominators bounded. We give explicitly the denominator in terms of Bernoulli numbers. This generalizes the role played by the rational function given by the difference of the Rademacher function and the classical Dedekind sums. We associate an exponential sum to the generalized Dedekind sums using the integrality of the generalized Rademacher function. We show that this exponential sum has a nontrivial bound that is sufficient to fulfill Weyl's equidistribution criterion and thus the fractional part of the generalized Dedekind sums are equidistributed. As an example, for a 3 dimensional case and Zagier's higher dimensional generalization of Dedekind sums, we compute the Laurent polynomials associated.

Motivation & Objective

• To generalize the classical Rademacher function to higher-dimensional Dedekind sums using Todd series and lattice cone decompositions.
• To establish a bounded denominator for generalized Dedekind sums by associating a Laurent polynomial that corrects for unbounded denominators.
• To prove the equidistribution of fractional parts of generalized Dedekind sums using nontrivial bounds on exponential sums derived from integrality of the generalized Rademacher function.
• To compute explicit Laurent polynomials for specific cases, such as 3-dimensional Dedekind sums and Zagier's higher-dimensional generalization.
• To provide a framework for understanding partial zeta values of totally real fields at nonpositive integers through generalized Dedekind sums and their arithmetic properties.

Proposed method

• Uses Todd series of lattice cones to represent generalized Dedekind sums as coefficients, leveraging the cocycle property of Todd series for reciprocity laws.
• Applies the iterated residue formula to decompose Todd series coefficients for nonsingular lattice cone decompositions, enabling explicit computation.
• Defines a generalized Rademacher function as a Laurent polynomial added to generalized Dedekind sums to bound their denominators, with the denominator explicitly expressed in terms of Bernoulli numbers.
• Constructs an exponential sum from the integrality of the generalized Rademacher function and derives a nontrivial bound sufficient to satisfy Weyl’s equidistribution criterion.
• Employs Parshin points and iterated constant terms (residues) in formal power series rings to define the generalized Rademacher function, with order-dependent behavior under variable reordering.
• Utilizes the iterated constant term operator $\operatorname{iCT}_{\mathfrak{A}}$ defined via successive residue maps to extract coefficients from rational functions, ensuring $\mathfrak{A}$-admissibility under specific algebraic conditions on the denominator.

Experimental results

Research questions

• RQ1 How can the classical Rademacher function be generalized to higher-dimensional Dedekind sums in $n$ dimensions?
• RQ2 What is the structure of the denominator of generalized Dedekind sums, and can it be uniformly bounded using a correction term?
• RQ3 Does the fractional part of generalized Dedekind sums equidistribute modulo 1, and what conditions ensure this?
• RQ4 How can the Todd series of lattice cones be used to compute generalized Dedekind sums and their reciprocity laws?
• RQ5 What explicit Laurent polynomials arise in the correction of generalized Dedekind sums for specific cases, such as 3-dimensional or Zagier-type sums?

Key findings

• The generalized Rademacher function is constructed as a Laurent polynomial that, when added to a generalized Dedekind sum of fixed index, ensures the denominator is bounded and explicitly expressed in terms of Bernoulli numbers.
• The fractional parts of generalized Dedekind sums are equidistributed modulo 1, as shown by deriving a nontrivial bound on the associated exponential sum that satisfies Weyl’s equidistribution criterion.
• For the 3-dimensional case and Zagier’s higher-dimensional generalization, the paper explicitly computes the Laurent polynomials associated with the generalized Rademacher function.
• The iterated residue formula allows for the decomposition of Todd series coefficients corresponding to nonsingular lattice cone decompositions, enabling precise computation of generalized Dedekind sums.
• The generalized Rademacher function is integral, and this integrality is used to construct an exponential sum with a nontrivial bound, which is essential for proving equidistribution.
• The method of iterated constant terms via Parshin points provides a systematic way to define and compute coefficients of rational functions in multiple variables, with order-dependent results that reflect the geometry of the lattice cone.