Generalized Dedekind Sums Overview

• Generalized Dedekind sums are arithmetic functions that extend the classical sum to various settings using cusp sets, Dirichlet characters, and Todd series, encapsulating key reciprocity and symmetry properties.
• They are constructed through diverse methods such as Eisenstein–Kronecker series, period integrals, and Bernoulli-type extensions, enabling precise evaluation and analytic insights within modular frameworks.
• These sums underpin advanced applications including equidistribution theorems, lattice computations, and quantum modular form analyses, providing actionable insights in modern number theory.

Generalized Dedekind sums are arithmetic functions modeled on the classical Dedekind sum $s(h,k)$, but defined in several distinct and structurally parallel settings: on cusp sets of non-cocompact lattices, on congruence subgroups with Dirichlet characters, as coefficients of Todd series of lattice cones, on complex lattices through Eisenstein–Kronecker series, and via period integrals or special $L$-values of modular forms. The literature does not single out one universal extension; instead, it exhibits a family of theories sharing a common architecture of reciprocity laws, cocycle or crossed-homomorphism identities, transformation formulas, equidistribution or density results, and arithmetic constraints on images and denominators (Burrin, 2015, Lee et al., 2016, Berkopec et al., 2022, Murakami, 18 Feb 2026).

1. Classical prototype and persistent structural features

The prototype is the classical Dedekind sum

$$s(h,k)=\sum_{n=1}^{k-1}\Bigl(\!\bigl(\tfrac{n}{k}\bigr)\Bigr)\,\Bigl(\!\bigl(\tfrac{hn}{k}\bigr)\Bigr), \qquad ((x))=x-\lfloor x\rfloor-\tfrac12,$$

for $\gcd(h,k)=1$ and $k>0$. It also has the cotangent representation

$$s(h,k)=\frac1{4k}\sum_{m=1}^{k-1}\cot\!\Bigl(\pi\frac{m}{k}\Bigr)\,\cot\!\Bigl(\pi\frac{hm}{k}\Bigr),$$

and it satisfies the reciprocity law

$$s(h,k)+s(k,h) = -\tfrac14+\tfrac1{12}\Bigl(\tfrac{h}{k}+\tfrac1{hk}+\tfrac{k}{h}\Bigr).$$

These formulas already display the two themes that recur throughout the generalized theory: finite arithmetic sums built from Bernoulli or sawtooth data, and reciprocity identities expressing a hidden symmetry between the parameters (Burrin, 2015).

The classical sum arises in the transformation law of the Dedekind $\eta$-function and can be reformulated on the cusp set $\mathbb Q\cup\{\infty\}$ of $SL_2(\mathbb Z)$. It also exhibits strong distribution behavior: the graph $L$0 is dense in $L$1, hence $L$2 is dense in $L$3, and the values are uniformly distributed on the unit interval modulo $L$4 (Berkopec et al., 2022, Burrin, 2015).

Average-size questions are likewise already present in the classical case. Walum’s exact second-moment formula expresses $L$5 in terms of fourth moments of $L$6 over odd Dirichlet characters modulo $L$7, which foreshadows the $L$8-value formulas that appear in several modern generalizations (Dillon et al., 2019).

2. Modular-cusp symbols and character-valued newform sums

One modular generalization replaces the integer pair $L$9 by a double coset in a non-cocompact lattice $$s(h,k)=\sum_{n=1}^{k-1}\Bigl(\!\bigl(\tfrac{n}{k}\bigr)\Bigr)\,\Bigl(\!\bigl(\tfrac{hn}{k}\bigr)\Bigr), \qquad ((x))=x-\lfloor x\rfloor-\tfrac12,$$0. Fix a cusp $$s(h,k)=\sum_{n=1}^{k-1}\Bigl(\!\bigl(\tfrac{n}{k}\bigr)\Bigr)\,\Bigl(\!\bigl(\tfrac{hn}{k}\bigr)\Bigr), \qquad ((x))=x-\lfloor x\rfloor-\tfrac12,$$1, choose a scaling matrix $$s(h,k)=\sum_{n=1}^{k-1}\Bigl(\!\bigl(\tfrac{n}{k}\bigr)\Bigr)\,\Bigl(\!\bigl(\tfrac{hn}{k}\bigr)\Bigr), \qquad ((x))=x-\lfloor x\rfloor-\tfrac12,$$2, and write $$s(h,k)=\sum_{n=1}^{k-1}\Bigl(\!\bigl(\tfrac{n}{k}\bigr)\Bigr)\,\Bigl(\!\bigl(\tfrac{hn}{k}\bigr)\Bigr), \qquad ((x))=x-\lfloor x\rfloor-\tfrac12,$$3. The Eisenstein series at $$s(h,k)=\sum_{n=1}^{k-1}\Bigl(\!\bigl(\tfrac{n}{k}\bigr)\Bigr)\,\Bigl(\!\bigl(\tfrac{hn}{k}\bigr)\Bigr), \qquad ((x))=x-\lfloor x\rfloor-\tfrac12,$$4,

$$s(h,k)=\sum_{n=1}^{k-1}\Bigl(\!\bigl(\tfrac{n}{k}\bigr)\Bigr)\,\Bigl(\!\bigl(\tfrac{hn}{k}\bigr)\Bigr), \qquad ((x))=x-\lfloor x\rfloor-\tfrac12,$$5

has a Laurent expansion at $$s(h,k)=\sum_{n=1}^{k-1}\Bigl(\!\bigl(\tfrac{n}{k}\bigr)\Bigr)\,\Bigl(\!\bigl(\tfrac{hn}{k}\bigr)\Bigr), \qquad ((x))=x-\lfloor x\rfloor-\tfrac12,$$6. Its Kronecker limit function produces a harmonic function $$s(h,k)=\sum_{n=1}^{k-1}\Bigl(\!\bigl(\tfrac{n}{k}\bigr)\Bigr)\,\Bigl(\!\bigl(\tfrac{hn}{k}\bigr)\Bigr), \qquad ((x))=x-\lfloor x\rfloor-\tfrac12,$$7, then a holomorphic $$s(h,k)=\sum_{n=1}^{k-1}\Bigl(\!\bigl(\tfrac{n}{k}\bigr)\Bigr)\,\Bigl(\!\bigl(\tfrac{hn}{k}\bigr)\Bigr), \qquad ((x))=x-\lfloor x\rfloor-\tfrac12,$$8 with $$s(h,k)=\sum_{n=1}^{k-1}\Bigl(\!\bigl(\tfrac{n}{k}\bigr)\Bigr)\,\Bigl(\!\bigl(\tfrac{hn}{k}\bigr)\Bigr), \qquad ((x))=x-\lfloor x\rfloor-\tfrac12,$$9, and from this a real-valued 1-cocycle

$\gcd(h,k)=1$0

The associated generalized Dedekind symbol is

$\gcd(h,k)=1$1

defined on $\gcd(h,k)=1$2. For $\gcd(h,k)=1$3, this construction recovers $\gcd(h,k)=1$4 (Burrin, 2015).

A different, but closely related, family is attached to primitive Dirichlet characters $\gcd(h,k)=1$5 and $\gcd(h,k)=1$6 satisfying $\gcd(h,k)=1$7. For $\gcd(h,k)=1$8 with $\gcd(h,k)=1$9, the newform Dedekind sum is

$k>0$0

where $k>0$1 for $k>0$2 and $k>0$3 otherwise. This sum depends only on $k>0$4 and satisfies a crossed-homomorphism relation on $k>0$5, becoming an ordinary homomorphism on $k>0$6 (Dillon et al., 2019).

Majure proved that the set of values of $k>0$7 on $k>0$8 is a free $k>0$9-module of rank $$s(h,k)=\frac1{4k}\sum_{m=1}^{k-1}\cot\!\Bigl(\pi\frac{m}{k}\Bigr)\,\cot\!\Bigl(\pi\frac{hm}{k}\Bigr),$$0, hence a full-rank lattice in the number field generated by the character values. The same theory yields a generalized Knopp identity,

$$s(h,k)=\frac1{4k}\sum_{m=1}^{k-1}\cot\!\Bigl(\pi\frac{m}{k}\Bigr)\,\cot\!\Bigl(\pi\frac{hm}{k}\Bigr),$$1

which reduces to Knopp’s classical identity when the characters are trivial (Majure, 2022).

3. Todd series, higher-dimensional cones, and Bernoulli-type extensions

A major branch of the theory interprets generalized Dedekind sums as coefficients of Todd series. For $$s(h,k)=\frac1{4k}\sum_{m=1}^{k-1}\cot\!\Bigl(\pi\frac{m}{k}\Bigr)\,\cot\!\Bigl(\pi\frac{hm}{k}\Bigr),$$2 with $$s(h,k)=\frac1{4k}\sum_{m=1}^{k-1}\cot\!\Bigl(\pi\frac{m}{k}\Bigr)\,\cot\!\Bigl(\pi\frac{hm}{k}\Bigr),$$3, let $$s(h,k)=\frac1{4k}\sum_{m=1}^{k-1}\cot\!\Bigl(\pi\frac{m}{k}\Bigr)\,\cot\!\Bigl(\pi\frac{hm}{k}\Bigr),$$4 be the cone spanned by $$s(h,k)=\frac1{4k}\sum_{m=1}^{k-1}\cot\!\Bigl(\pi\frac{m}{k}\Bigr)\,\cot\!\Bigl(\pi\frac{hm}{k}\Bigr),$$5 and $$s(h,k)=\frac1{4k}\sum_{m=1}^{k-1}\cot\!\Bigl(\pi\frac{m}{k}\Bigr)\,\cot\!\Bigl(\pi\frac{hm}{k}\Bigr),$$6. Writing

$$s(h,k)=\frac1{4k}\sum_{m=1}^{k-1}\cot\!\Bigl(\pi\frac{m}{k}\Bigr)\,\cot\!\Bigl(\pi\frac{hm}{k}\Bigr),$$7

one defines generalized Dedekind sums $$s(h,k)=\frac1{4k}\sum_{m=1}^{k-1}\cot\!\Bigl(\pi\frac{m}{k}\Bigr)\,\cot\!\Bigl(\pi\frac{hm}{k}\Bigr),$$8 by

$$s(h,k)=\frac1{4k}\sum_{m=1}^{k-1}\cot\!\Bigl(\pi\frac{m}{k}\Bigr)\,\cot\!\Bigl(\pi\frac{hm}{k}\Bigr),$$9

For $$s(h,k)+s(k,h) = -\tfrac14+\tfrac1{12}\Bigl(\tfrac{h}{k}+\tfrac1{hk}+\tfrac{k}{h}\Bigr).$$0, this is the classical Dedekind sum. Lee–Jun–Chae derived a higher Hickerson formula expressing $$s(h,k)+s(k,h) = -\tfrac14+\tfrac1{12}\Bigl(\tfrac{h}{k}+\tfrac1{hk}+\tfrac{k}{h}\Bigr).$$1 in terms of the continued fraction of $$s(h,k)+s(k,h) = -\tfrac14+\tfrac1{12}\Bigl(\tfrac{h}{k}+\tfrac1{hk}+\tfrac{k}{h}\Bigr).$$2, splitting it into an integral part $$s(h,k)+s(k,h) = -\tfrac14+\tfrac1{12}\Bigl(\tfrac{h}{k}+\tfrac1{hk}+\tfrac{k}{h}\Bigr).$$3 and a fractional part $$s(h,k)+s(k,h) = -\tfrac14+\tfrac1{12}\Bigl(\tfrac{h}{k}+\tfrac1{hk}+\tfrac{k}{h}\Bigr).$$4. Substituting this decomposition into Siegel’s formula for partial zeta-values yields a higher Meyer formula involving only the integral part (Lee et al., 2016).

The higher-dimensional version, due to Chae–Jun–Lee, fixes an index $$s(h,k)+s(k,h) = -\tfrac14+\tfrac1{12}\Bigl(\tfrac{h}{k}+\tfrac1{hk}+\tfrac{k}{h}\Bigr).$$5 and defines

$$s(h,k)+s(k,h) = -\tfrac14+\tfrac1{12}\Bigl(\tfrac{h}{k}+\tfrac1{hk}+\tfrac{k}{h}\Bigr).$$6

with periodic Bernoulli functions $$s(h,k)+s(k,h) = -\tfrac14+\tfrac1{12}\Bigl(\tfrac{h}{k}+\tfrac1{hk}+\tfrac{k}{h}\Bigr).$$7. These sums are again coefficients of Todd series, and their reciprocity laws come from the cocycle property of lattice cones. A nonsingular decomposition together with iterated residues produces a correcting Laurent polynomial and an integral generalized Rademacher function controlling the denominators of the sums (Chae et al., 2014).

Beck–Chavez introduced Bernoulli–Dedekind sums

$$s(h,k)+s(k,h) = -\tfrac14+\tfrac1{12}\Bigl(\tfrac{h}{k}+\tfrac1{hk}+\tfrac{k}{h}\Bigr).$$8

which, in their formulation, generalize and unify sums introduced by Dedekind, Apostol, Carlitz, Rademacher, Sczech, Hall–Wilson–Zagier, and others. Their reciprocity theorem is proved by a combinatorial generating-function argument using Raabe’s formula and a comparison of sign matrices. Brown’s floor-sum formulas

$$s(h,k)+s(k,h) = -\tfrac14+\tfrac1{12}\Bigl(\tfrac{h}{k}+\tfrac1{hk}+\tfrac{k}{h}\Bigr).$$9

show, in turn, that each $\eta$0 can be written as a linear combination of higher-order Dedekind-type sums

$\eta$1

linking floor-function identities directly to generalized Dedekind structures (Beck et al., 2010, Brown, 15 Jul 2025).

4. Reciprocity laws, trigonometric forms, and distribution phenomena

Reciprocity remains the defining invariant across most variants. Periodic analogues built from periodic Bernoulli functions $\eta$2 occur in transformation formulas for generalized Eisenstein series. In this setting, the periodic Dedekind sum $\eta$3 appears as the obstruction to modular invariance at $\eta$4 and satisfies an explicit reciprocity theorem, together with a shifted two-variable generalization $\eta$5 (Dağlı et al., 2015).

Fourier–Dedekind sums form another distinct generalization. For integers $\eta$6 coprime to $\eta$7,

$\eta$8

These sums admit a convolution factorization into one-dimensional factors, generate an abelian group under convolution, and possess reduced versions with generating-function and geometric descriptions. Tsukerman also extends Rademacher reciprocity to a larger range of the spectral parameter $\eta$9, proves average-behavior theorems, and establishes sharp upper and lower bounds in the two-dimensional case (Tsukerman, 2013).

Discrete Fourier transform methods provide uniform trigonometric representations. Rassias–Tóth show that the higher-dimensional Dedekind–Bernoulli sum $\mathbb Q\cup\{\infty\}$0 can be written, when $\mathbb Q\cup\{\infty\}$1 is even, as a finite sum of products of cotangent derivatives. He–Shi derive reciprocity formulas for generalized Dedekind–Rademacher sums from an exact product formula for Bernoulli functions. Kim–Kim–Lee–Jang define poly-Dedekind sums $\mathbb Q\cup\{\infty\}$2 by replacing one Bernoulli factor with a type 2 poly-Bernoulli function, while Ma–Kim–Lee–Kim–Kim introduce poly-Dedekind-type DC sums $\mathbb Q\cup\{\infty\}$3 built from poly-Euler functions. Each of these constructions comes with a reciprocity law (Rassias et al., 2015, He et al., 2023, Kim et al., 2020, Ma et al., 2020).

Distribution theorems are equally persistent. For a non-cocompact lattice $\mathbb Q\cup\{\infty\}$4 and any real $\mathbb Q\cup\{\infty\}$5, the values $\mathbb Q\cup\{\infty\}$6, ordered by the lower-left entry $\mathbb Q\cup\{\infty\}$7, are equidistributed in $\mathbb Q\cup\{\infty\}$8. The proof uses a Vardi-type identity expressing exponential sums of Dedekind symbols through twisted Kloosterman sums, together with Goldfeld–Sarnak bounds and Weyl’s criterion. For the Todd-series sums $\mathbb Q\cup\{\infty\}$9, Jun–Lee attach exponential sums and use Denef–Loeser purity to prove Weil-type bounds, implying equidistribution of the fractional parts of $SL_2(\mathbb Z)$0. Lee–Jun–Chae obtain an analogous equidistribution theorem for the fractional-part graph $SL_2(\mathbb Z)$1 (Burrin, 2015, Jun et al., 2013, Lee et al., 2016).

5. Arithmetic image, moments, and computation

Analytic questions about size and average behavior have been studied particularly thoroughly for the two-character newform family. Dillon–Gaston compute the finite Fourier transform

$SL_2(\mathbb Z)$2

and obtain an exact second-moment formula in terms of $SL_2(\mathbb Z)$3-values and a Gauss-sum convolution $SL_2(\mathbb Z)$4. Their upper and lower bounds,

$SL_2(\mathbb Z)$5

imply the asymptotic growth law

$SL_2(\mathbb Z)$6

This extends Walum’s classical second-moment identity to a two-character setting (Dillon et al., 2019).

Arithmetic image problems lead to denominator bounds and lattice-containment questions. Knight–Matos–Sefidi–Young prove that for every $SL_2(\mathbb Z)$7,

$SL_2(\mathbb Z)$8

When $SL_2(\mathbb Z)$9 and $L$00 are primitive quadratic of odd conductors $L$01, this yields integrality if $L$02. They formulate the Two-Conjecture, predicting

$L$03

in the quadratic case, and a generalized version when the character field is quadratic (Knight et al., 12 Mar 2025).

Computationally, Tranbarger–Wang replace the defining double sum with a group-theoretic rewriting algorithm. After an $L$04 precomputation for $L$05, their method computes $L$06 in $L$07 time by combining a TS-decomposition in $L$08, modified Reidemeister rewriting, and a finite set of Schreier generators. In a different asymptotic direction, Borda studies inhomogeneous sums

$L$09

reducing $L$10 to a Diophantine sum $L$11 and then to an asymptotic description in the continued-fraction partial quotients of $L$12 (Tranbarger et al., 2022, Borda, 2016).

6. Elliptic, higher-weight, and quantum-modular extensions

Sczech’s elliptic Dedekind sums transfer the theory from the modular group to complex lattices. If $L$13 has multiplier ring $L$14 equal to an order in an imaginary quadratic field, then

$L$15

where $L$16 is the analytic-continuation value of an Eisenstein–Kronecker series. With Ito’s normalization

$L$17

one has $L$18 when the $L$19-invariant of $L$20 is real. Under the exclusions

$L$21

the set of normalized values $L$22 is dense in $L$23 (Berkopec et al., 2022).

Higher-weight newform sums arise from period integrals of holomorphic Eisenstein series. For primitive nontrivial Dirichlet characters $L$24 with $L$25, one defines

$L$26

and

$L$27

for $L$28 with $L$29. These sums admit a finite-sum formula involving $L$30, satisfy a Fricke reciprocity law, and define a weight $L$31 quantum modular form on $L$32: the cocycle error extends to a polynomial of degree $L$33. After the further normalization $L$34, Sage–Math computations for quadratic character pairs with $L$35 and $L$36 support a divisibility conjecture for the image (Tranbarger, 19 Dec 2025).

Murakami gives a still broader extension attached to any modular form $L$37 of weight $L$38 on a congruence subgroup $L$39, defining

$L$40

For Eisenstein series $L$41, these values have a single trigonometric-sum expression involving $L$42 and $L$43. They satisfy a quantum-modular transformation law in which the modular defect is given by an explicit period function plus a correction term, and they arise as radial limits of Eichler integrals. In the genus-$L$44 $L$45-TQFT case, the same trigonometric expressions coincide with the signature formula and imply its quantum modularity (Murakami, 18 Feb 2026).