Fourier-Dedekind Sums

• Fourier-Dedekind sums are multidimensional generalizations of Dedekind sums defined via finite trigonometric or exponential sums that unify several classical variants.
• They exhibit a rich convolution structure, deep algebraic-analytic properties, and satisfy generalized reciprocity laws linked to modular forms and Eisenstein series.
• Their computation leverages modern algorithms like Barvinok’s cone decomposition, enabling effective evaluation in lattice-point enumeration and quasi-polynomial analysis.

Fourier-Dedekind sums are multidimensional generalizations of classical Dedekind sums, intimately associated with lattice-point enumeration, modular forms, convex polytopes, transformation properties of Eisenstein series, and analytic number theory. Defined via finite trigonometric or exponential sums, these objects exhibit highly structured algebraic, analytic, and geometric properties, and unify several disparate generalizations—such as character-Dedekind, Hardy–Berndt, and periodic Bernoulli sums—through the abstract lens of discrete Fourier analysis.

1. Definitions and Structural Framework

Let $b>1$ and $a_1,\dots,a_d\in\mathbb{Z}$, with each $a_j$ coprime to $b$. The $d$-dimensional Fourier-Dedekind sum is defined by

$$S_{(a_1,\dots,a_d;b)}(t)= \frac{1}{b}\sum_{j=1}^{b-1} \frac{\exp\left(\frac{2\pi i}{b}jt\right)} {(1-\exp(2\pi i j a_1/b))\cdots(1-\exp(2\pi i j a_d/b))},\quad t\in\mathbb{Z}.$$

Alternatively, these sums can be formulated via finite difference equations: $S_{(a_1,\ldots,a_d;b)}(t)$ is the unique zero-mean, $b$-periodic function $f$ satisfying $(I-T^{a_1})\cdots(I-T^{a_d})f=S_b$, where $T^a f(t)=f(t+a)$ and $S_b(t)=\delta_b(t)-1/b$.

Fourier-Dedekind sums enjoy a rich convolution structure: they are characterized as the convolution $$S_{(a_1,\dots,a_d;b)}=S_{(a_1;b)}*\cdots*S_{(a_d;b)}$$, with convolution defined on the space of $b$-periodic, zero-mean functions. The set of such sums forms an abelian group under convolution, which can be built from the $1$-dimensional pieces and finite difference operators. For $d\geq1$, there are explicit integer-linear decompositions expressing higher-dimensional sums as sums of lower-dimensional ones (Tsukerman, 2013).

Generalized forms include inclusion of Dirichlet characters and periodic Bernoulli functions. For primitive Dirichlet characters $\chi_1$ (mod $q_1$), $\chi_2$ (mod $q_2$) with $\chi_1\chi_2(-1)=1$, let $B_1(x)$ denote the sawtooth or first Bernoulli function. Then

$$S_{\chi_1,\chi_2}(a,c) = \sum_{j\bmod c} \sum_{n\bmod q_1} \overline{\chi_2}(j)\,\overline{\chi_1}(n) \,B_1\left(\frac{j}{c}\right) B_1\left(\frac{n}{q_1}+\frac{a j}{c}\right),$$

recovering the classical Dedekind sum when both characters are trivial (Dillon et al., 2019).

2. Analytic and Algebraic Properties

Fourier-Dedekind sums are governed by deep algebraic and analytic structures. The finite Fourier transform in the $a$ variable,

$$\widehat{S}_{(a_1,\dots,a_d;b)}(\xi) = \sum_{\substack{a\bmod b\ (a,b)=1}} S_{(a_1,\dots,a_d;b)}(a)\,\xi(a),$$

for $\xi$ a Dirichlet character modulo $b$, admits explicit trigonometric or $L$-function expressions in terms of cotangent products, higher derivatives, or twisted $L$-values, depending on context and generalization (Rassias et al., 2015, Dillon et al., 2019).

Parseval-type identities relate mean-square values (second moments) of these sums to explicit spectral transforms. For the character-summed setting, the main formula is:

$\sum_{\substack{a\bmod c\(a,c)=1}} |S_{\chi_1,\chi_2}(a,c)|^2 = \frac1{\varphi(c)} \sum_{\xi\bmod c} |\widehat S_{\chi_1,\chi_2}(\xi)|^2,$

with closed forms available in terms of $L$-functions and weighted arithmetic convolutions (Dillon et al., 2019).

The sums are intimately related to generating functions for counting lattice points in rational polytopes. For $s_n(a_1,\ldots,a_d; b)$,

$$s_n(a_1,\ldots,a_d; b) = \mathrm{CT}_\lambda \left[ \frac{\lambda^n}{(1-\lambda^b)\prod_{i=1}^d (1-\lambda^{a_i})} \right] - \frac{1}{b\prod_{i=1}^d(1-1)},$$

where $\mathrm{CT}_\lambda$ denotes the constant term in $\lambda$ (Xin et al., 2023).

3. Reciprocity Laws and Functional Equations

Fourier-Dedekind sums generalize the celebrated Rademacher and Dedekind reciprocity laws. For pairwise coprime $a_1,\dots,a_d>0$, define

$$R_{\{a_1,\dots,a_d\}}(n) = \sum_{m=1}^d S_{(a_1,\dots,\widehat{a_m},\dots,a_d;a_m)}(n),$$

with classical Rademacher reciprocity stating

$$R_{\{a_1,\dots,a_d\}}(n) = -\mathrm{poly}_{\{a_1,\dots,a_d\}}(-n),$$

for $1 \leq n \leq a_1+\cdots+a_d-1$; here, $\mathrm{poly}_{\{a\}}(t)$ denotes the polynomial part of the restricted partition function. The reciprocity extends to a wider range of $n$, and similar theorems exist in the periodic and character-twisted cases, with explicit Bernoulli and partition-theoretic corrections (Tsukerman, 2013, Dağlı et al., 2015).

Functional equations provide finite-difference lowering:

$$(I-T^{a_d}) S_{(a_1,\dots,a_d;b)} = S_{(a_1,\dots,a_{d-1};b)},$$

and, in the Eisenstein series and periodic Bernoulli context, the modular transformation formula features Fourier-Dedekind sums as obstruction terms to invariance.

4. Fourier Analysis and Trigonometric Formulas

The discrete Fourier analytic approach, including multilinear Parseval identities and explicit DFT-pairs for periodic Bernoulli and sawtooth functions, enables uniform derivation of all classical and higher-dimensional Dedekind-type sums, Hardy sums, and their cotangent/trigonometric analogs (Rassias et al., 2015).

Key trigonometric representation:

$$s(h,k) = -\frac{1}{4k}\sum_{a=1}^{k-1} \cot\left(\frac{\pi a}{k}\right) \cot\left(\frac{\pi a h}{k}\right)$$

generalizes to explicit finite sums for higher dimensions, with cotangent products and derivatives encoding the combinatorial complexity. The entire family (involving Hardy-Berndt, periodic zeta, and Bernoulli sums) is reducible to discrete Fourier transforms and a handful of kernel identities.

For sums $S_{r_1,\dots,r_m}(h_1,\dots,h_m;k)$ involving Bernoulli functions,

$$S_{r_1,\ldots,r_m}(h_1,\ldots,h_m;k) = (-1)^{(r_1+\ldots+r_m)/2} \frac{(2\pi)^{r_1+\ldots+r_m}}{2\,k^{r_1+\ldots+r_m-1}\prod_j r_j!} \sum_{a=1}^{k-1} \prod_{j=1}^m \cot^{(r_j-1)}\left(\frac{\pi a h_j}{k}\right)$$

(Rassias et al., 2015).

5. Computation and Algorithmic Developments

Fourier-Dedekind sums, particularly in high dimension, are nontrivial to compute via the definition, as the complexity grows rapidly. The modular Barvinok simplicial cone decomposition algorithm provides a polynomial-time solution in fixed dimension, reducing evaluation to constant-term computation over a rational generating function decomposed into short rational summands (Xin et al., 2023).

Summary of algorithmic steps:

1. To compute $s_n(a_1,\ldots,a_d; b)$, write the generating function as a rational function of auxiliary variables.
2. Apply Barvinok's cone decomposition via, e.g., LattE, to express as sum of rational functions with unimodular denominators.
3. Extract the constant term successively in each variable, yielding the Fourier-Dedekind sum value.

For $d$ fixed, overall cost is polynomial in bit-length of inputs; this is practical for applications to Ehrhart theory and quasi-polynomiality of lattice-point enumerators.

6. Applications, Extremal Behavior, and Further Developments

Fourier-Dedekind sums arise naturally in the Ehrhart theory of lattice polytopes, syndrome defects in topology, modular transformations of Eisenstein series, and equidistribution problems in analytic number theory. Notably, (Jun et al., 2013) shows that for generalized Dedekind sums $s_{ij}(p,q)$ defined as coefficients of the Todd series of a planar cone, the Weyl equidistribution criterion is satisfied, based on exponential sum bounds via Denef–Loeser’s theorem on the purity of $l$-adic cohomology. The classical case (with scaling factor $12$ for $s(p,q)$) is subsumed as a first explicit instance.

Extremal values in the two-dimensional setting are precisely bounded: for $\gcd(a_1,a_2,b)=1$,

$$-\frac{(b-1)(b-5)}{12b} \leq S_{(a_1,a_2;b)}(0) \leq \frac{(b-1)(b+1)}{12b},$$

with similar explicit characterizations for maximum and minimum locations (Tsukerman, 2013).

New directions include:

• Precise asymptotics and expansion error elimination in mean-square estimates (Dillon et al., 2019),
• Moments and higher Fourier statistics,
• Translation to spectral theory of modular forms,
• Analytical and diophantine applications via transformation law obstructions and quasi-modularity,
• Arithmetic geometry and equidistribution on toric and modular surfaces.

7. Connections with Modular Forms and Generalized Eisenstein Series

Periodic analogues of Dedekind sums, formed via periodic Bernoulli functions attached to arbitrary periodic sequences or Dirichlet characters, appear as noninvariant terms in modular or automorphic transformations of generalized Eisenstein series. Explicit transformation and reciprocity laws control the finer arithmetic decompositions of these modular objects. In particular, $s(d,c;B_b,A_c)$ and its reciprocity formula unify earlier character, alternating, and Hardy-Berndt sum variants (Dağlı et al., 2015).

Special cases recover classical results, such as:

• Character-Dedekind reciprocity: For primitive characters $X_1, X_2$,

$$s(c,d;X_{1,c},X_{2,b}) + s(d,c;X_{2,b},X_{1,c}) = -B_1(X_1)B_1(X_2),$$

• Infinite series identities via evaluation of Eisenstein coefficients, demonstrated to include Ramanujan-type and Cauchy formulas as corollaries.

In summary, the theory of Fourier-Dedekind sums subsumes the main structural, analytic, and computational principles underlying Dedekind-type phenomena in modern number theory and discrete geometry, providing a canonical framework for generalization and further exploration (Tsukerman, 2013, Rassias et al., 2015, Xin et al., 2023, Jun et al., 2013, Dağlı et al., 2015, Dillon et al., 2019).