---
title: Generalized Dedekind Sums Overview
url: https://www.emergentmind.com/topics/generalized-dedekind-sums
type: topic
---

# Generalized Dedekind Sums Overview

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 [1509.04429] [1608.02472] [2208.03286] [2602.16159].

## 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 [1509.04429].

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 \(\{(h/k,s(h,k))\}\) is dense in \(\mathbb R^2\), hence \(\{s(h,k)\}\) is dense in \(\mathbb R\), and the values are uniformly distributed on the unit interval modulo \(1\) [2208.03286] [1509.04429].

Average-size questions are likewise already present in the classical case. Walum’s exact second-moment formula expresses \(\sum_{a\bmod p}|s(a,p)|^2\) in terms of fourth moments of \(L(1,\psi)\) over odd Dirichlet characters modulo \(p\), which foreshadows the \(L\)-value formulas that appear in several modern generalizations [1907.11745].

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

One modular generalization replaces the integer pair \((h,k)\) by a double coset in a non-cocompact lattice \(\Gamma<SL_2(\mathbb R)\). Fix a cusp \(a\), choose a scaling matrix \(\sigma_a\), and write \(\Gamma^a=\sigma_a^{-1}\Gamma\sigma_a\). The Eisenstein series at \(\infty\),
\[
E(z,s)=\sum_{\gamma\in\Gamma^a_\infty\backslash\Gamma^a}\Im(\gamma z)^s,
\]
has a Laurent expansion at \(s=1\). Its Kronecker limit function produces a harmonic function \(H(z)\), then a holomorphic \(F(z)\) with \(\Re F(z)=H(z)\), and from this a real-valued 1-cocycle
\[
\phi_a(\gamma)=\frac{1}{2\pi i}\bigl(F(\gamma z)-F(z)-2\log(cz+d)\bigr).
\]
The associated generalized Dedekind symbol is
\[
S_a([\gamma])=\frac{V}{4\pi}\frac{a+d}{c}-\phi_a(\gamma),
\]
defined on \(\Gamma_\infty\backslash\Gamma^a/\Gamma_\infty\). For \(\Gamma=SL_2(\mathbb Z)\), this construction recovers \(\operatorname{sign}(c)\,s(a;|c|)\) [1509.04429].

A different, but closely related, family is attached to primitive Dirichlet characters \(\chi_1\bmod q_1\) and \(\chi_2\bmod q_2\) satisfying \(\chi_1\chi_2(-1)=1\). For
\(\gamma=\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in\Gamma_0(q_1q_2)\) with \(c\ge1\), the newform Dedekind sum is
\[
S_{\chi_1,\chi_2}(a,c)
=
\sum_{j\bmod c}\sum_{n\bmod q_1}
\overline\chi_2(j)\,\overline\chi_1(n)\,
B_1\!\Bigl(\tfrac{j}{c}\Bigr)\,
B_1\!\Bigl(\tfrac{n}{q_1}+\tfrac{aj}{c}\Bigr),
\]
where \(B_1(x)=0\) for \(x\in\mathbb Z\) and \(B_1(x)=x-\lfloor x\rfloor-\tfrac12\) otherwise. This sum depends only on \((a,c)\) and satisfies a crossed-homomorphism relation on \(\Gamma_0(q_1q_2)\), becoming an ordinary homomorphism on \(\Gamma_1(q_1q_2)\) [1907.11745].

Majure proved that the set of values of \(S_{\chi_1,\chi_2}\) on \(\Gamma_1(q_1q_2)\) is a free \(\mathbb Z\)-module of rank \([F_{\chi_1,\chi_2}:\mathbb Q]\), hence a full-rank lattice in the number field generated by the character values. The same theory yields a generalized Knopp identity,
\[
\sum_{\substack{ad=n\\(a,n)=1}}\psi(a)\sum_{b\bmod d}
S_{\chi_1,\chi_2}(ah+bk,dk)
=
\rho_{\chi_1,\chi_2}(n)\,S_{\chi_1,\chi_2}(h,k),
\]
which reduces to Knopp’s classical identity when the characters are trivial [2208.13060].

## 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 \(0<p<q\) with \(\gcd(p,q)=1\), let \(\sigma_{p,q}\subset\mathbb R^2\) be the cone spanned by \((1,0)\) and \((p,q)\). Writing
\[
\operatorname{Todd}_{p,q}(x,y)=\sum_{i,j\ge0} t_{i,j}(p,q)\,\frac{x^i y^j}{i!\,j!},
\]
one defines generalized Dedekind sums \(s_{i,j}(p,q)\) by
\[
t_{i,j}(p,q)
=
-(-q)^{i+j-1}\,[\,s_{i,j}(p,q)+\delta_{i,1}\delta_{j,1}B_iB_j\,].
\]
For \(i=j=1\), this is the classical Dedekind sum. Lee–Jun–Chae derived a higher Hickerson formula expressing \(s_{i,j}(p,q)\) in terms of the continued fraction of \(p/q\), splitting it into an integral part \(s^I_{i,j}(p,q)\) and a fractional part \(s^R_{i,j}(p,q)\). Substituting this decomposition into Siegel’s formula for partial zeta-values yields a higher Meyer formula involving only the integral part [1608.02472].

The higher-dimensional version, due to Chae–Jun–Lee, fixes an index \(\mathbf i=(i_1,\dots,i_n)\) and defines
\[
s_{i_1,\dots,i_n}(q;p_1,\dots,p_{n-1})
=
\sum_{k_1,\dots,k_{n-1}\bmod q}
\widetilde B_{i_1}\!\Bigl(\tfrac{k_1}{q}\Bigr)\cdots
\widetilde B_{i_{n-1}}\!\Bigl(\tfrac{k_{n-1}}{q}\Bigr)\,
\widetilde B_{i_n}\!\Bigl(\tfrac{\sum_{j=1}^{n-1}p_jk_j}{q}\Bigr),
\]
with periodic Bernoulli functions \(\widetilde B_j\). 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 [1406.3563].

Beck–Chavez introduced Bernoulli–Dedekind sums
\[
S_{p_1,\dots,p_n}(a_1,\dots,a_n;x_1,\dots,x_n)
=
\sum_{h\bmod a_k}\prod_{i\ne k}
\widetilde B_{p_i}\!\Bigl(a_i\frac{h+x_k}{a_k}-x_i\Bigr),
\]
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
\[
\mathbf S_r(n,m)=\sum_{k=1}^{n-1}\Bigl\lfloor \frac{km}{n}\Bigr\rfloor^r
\]
show, in turn, that each \(\mathbf S_r\) can be written as a linear combination of higher-order Dedekind-type sums
\[
\delta(b;a_1,\dots,a_n)=2^n b^{\,n-1}\sum_{k=1}^{b-1}\prod_{i=1}^n\Bigl\{\frac{k a_i}{b}\Bigr\},
\]
linking floor-function identities directly to generalized Dedekind structures [1008.0038] [2507.11666].

## 4. Reciprocity laws, trigonometric forms, and distribution phenomena

Reciprocity remains the defining invariant across most variants. Periodic analogues built from periodic Bernoulli functions \(P_m(x,A)\) occur in transformation formulas for generalized Eisenstein series. In this setting, the periodic Dedekind sum \(s(d,c;B_b,A_c)\) appears as the obstruction to modular invariance at \(s=0\) and satisfies an explicit reciprocity theorem, together with a shifted two-variable generalization \(s(d,c;A_b,A_c;x,y)\) [1506.01809].

Fourier–Dedekind sums form another distinct generalization. For integers \(a_1,\dots,a_d\) coprime to \(b\),
\[
S_{(a_1,\dots,a_d;\,b)}(t)
=
\frac1b\sum_{j=1}^{b-1}
\frac{\xi_b^{jt}}
{(1-\xi_b^{ja_1})\cdots(1-\xi_b^{ja_d})},
\qquad \xi_b=e^{2\pi i/b}.
\]
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 \(t\), proves average-behavior theorems, and establishes sharp upper and lower bounds in the two-dimensional case [1307.5542].

Discrete Fourier transform methods provide uniform trigonometric representations. Rassias–Tóth show that the higher-dimensional Dedekind–Bernoulli sum \(S_{r_1,\dots,r_m}(h_1,\dots,h_m;k)\) can be written, when \(r_1+\cdots+r_m\) 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 \(S_p^{(k)}(a,b)\) by replacing one Bernoulli factor with a type 2 poly-Bernoulli function, while Ma–Kim–Lee–Kim–Kim introduce poly-Dedekind-type DC sums \(T_p^{(k)}(h,m)\) built from poly-Euler functions. Each of these constructions comes with a reciprocity law [1512.01466] [2310.17093] [2009.04853] [2012.00264].

Distribution theorems are equally persistent. For a non-cocompact lattice \(\Gamma<SL_2(\mathbb R)\) and any real \(k>0\), the values \(k\,S_a([\gamma])\bmod 1\), ordered by the lower-left entry \(c([\gamma])\to\infty\), are equidistributed in \([0,1]\). 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 \(s_{i,j}(p,q)\), Jun–Lee attach exponential sums and use Denef–Loeser purity to prove Weil-type bounds, implying equidistribution of the fractional parts of \(R(i,j)q^{i+j-2}s_{i,j}(p,q)\). Lee–Jun–Chae obtain an analogous equidistribution theorem for the fractional-part graph \(\bigl(\frac pq,\{R_{i+j}q^{i+j-2}s_{i,j}(p,q)\}\bigr)\) [1509.04429] [1303.2331] [1608.02472].

## 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
\[
\widehat S_{\chi_1,\chi_2}(\xi)
=
\sum_{\substack{a\bmod c\\(a,c)=1}}
S_{\chi_1,\chi_2}(a,c)\,\xi(a),
\]
and obtain an exact second-moment formula in terms of \(L(1,\cdot)\)-values and a Gauss-sum convolution \(g_{\chi_1,\chi_2}(\psi;c)\). Their upper and lower bounds,
\[
\sum_{(a,c)=1}|S_{\chi_1,\chi_2}(a,c)|^2\ll_\varepsilon q_1c^{2+\varepsilon},
\qquad
\sum_{(a,c)=1}|S_{\chi_1,\chi_2}(a,c)|^2\gg_\varepsilon q_1c^{2-\varepsilon},
\]
imply the asymptotic growth law
\[
\sum_{(a,c)=1}|S_{\chi_1,\chi_2}(a,c)|^2=q_1c^{2+o(1)}
\quad (c\to\infty).
\]
This extends Walum’s classical second-moment identity to a two-character setting [1907.11745].

Arithmetic image problems lead to denominator bounds and lattice-containment questions. Knight–Matos–Sefidi–Young prove that for every \(\gamma\in\Gamma_1(q_1q_2)\),
\[
S_{\chi_1,\chi_2}(\gamma)\in \frac1{\gcd(q_1,q_2)}\,\mathbb Z[\chi_1,\chi_2].
\]
When \(\chi_1\) and \(\chi_2\) are primitive quadratic of odd conductors \(q_1,q_2>4\), this yields integrality if \(\gcd(q_1,q_2)=1\). They formulate the Two-Conjecture, predicting
\[
\operatorname{Im}S_{\chi_1,\chi_2}(\Gamma_1(q_1q_2))=2\mathbb Z
\]
in the quadratic case, and a generalized version when the character field is quadratic [2503.09741].

Computationally, Tranbarger–Wang replace the defining double sum with a group-theoretic rewriting algorithm. After an \(O(N^3)\) precomputation for \(N=q_1q_2\), their method computes \(S_{\chi_1,\chi_2}(\gamma)\) in \(O(\log|c|)\) time by combining a TS-decomposition in \(SL_2(\mathbb Z)\), modified Reidemeister rewriting, and a finite set of Schreier generators. In a different asymptotic direction, Borda studies inhomogeneous sums
\[
s_{p,q}(a,b)=\sum_{k=1}^{b-1}B_p\!\Bigl(\tfrac{k}{b}\Bigr)\,B_q\!\Bigl(\Bigl\{\tfrac{ak}{b}\Bigr\}\Bigr),
\]
reducing \(s_{p,p}(a,b)\) to a Diophantine sum \(\sum 1/(m^p\|ma/b\|^p)\) and then to an asymptotic description in the continued-fraction partial quotients of \(a/b\) [2210.01172] [1606.07946].

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

Sczech’s elliptic Dedekind sums transfer the theory from the modular group to complex lattices. If \(L\subset\mathbb C\) has multiplier ring \(\mathcal O_L\) equal to an order in an imaginary quadratic field, then
\[
D_L(h,k)
=
\frac1k\sum_{\mu\in L/kL}
E_1\!\Bigl(\tfrac{h\mu}{k}\Bigr)\,
E_1\!\Bigl(\tfrac{\mu}{k}\Bigr),
\]
where \(E_1\) is the analytic-continuation value of an Eisenstein–Kronecker series. With Ito’s normalization
\[
\widetilde D_L(h,k)
=
(i\sqrt{|d_L|}\,E_2(0))^{-1}D_L(h,k),
\]
one has \(\widetilde D_L(h,k)\in\mathbb R\) when the \(j\)-invariant of \(L\) is real. Under the exclusions
\[
\mathcal O_L\neq \mathbb Z,\ \mathbb Z[i],\ \mathbb Z[(1-\sqrt{-3})/2],
\]
the set of normalized values \(\{\widetilde D_L(a,c):c\neq0\}\) is dense in \(\mathbb R\) [2208.03286].

Higher-weight newform sums arise from period integrals of holomorphic Eisenstein series. For primitive nontrivial Dirichlet characters \(\chi_1,\chi_2\) with \(\chi_1\chi_2(-1)=(-1)^k\), one defines
\[
\phi_{\chi_1,\chi_2,k}(\gamma;X,Y)
=
\int_{\infty}^{\gamma\infty}
E_{\chi_1,\chi_2,k}(z)\,(Xz+Y)^{k-2}\,dz
\]
and
\[
S_{\chi_1,\chi_2,k}(\gamma)
=
(-1)^k\,\tau(\overline\chi_1)\,(k-1)\,
\phi_{\chi_1,\chi_2,k}\!\Bigl(\gamma;1,-\tfrac ac\Bigr),
\]
for \(\gamma=\begin{pmatrix}a&b\\ c&d\end{pmatrix}\) with \(c>0\). These sums admit a finite-sum formula involving \(B_{k-1}\), satisfy a Fricke reciprocity law, and define a weight \(2-k\) quantum modular form on \(\Gamma_1(N)\): the cocycle error extends to a polynomial of degree \(k-2\). After the further normalization \(\widetilde S(\gamma)=c^{k-2}S(\gamma)\), Sage–Math computations for quadratic character pairs with \(q_1q_2\le32\) and \(2\le k\le9\) support a divisibility conjecture for the image [2512.17139].

Murakami gives a still broader extension attached to any modular form \(f(\tau)=\sum_{n\ge0}a_n q^{n/N}\) of weight \(k\ge2\) on a congruence subgroup \(\Gamma\supset\Gamma(N)\), defining
\[
S_f\!\Bigl(\tfrac ac\Bigr)=\widehat L_f\!\Bigl(k-1;\tfrac ac\Bigr),
\qquad
\widehat L_f(s;x)=-(2\pi)^{-s}e(-s/4)\Gamma(s)L_f(s;x).
\]
For Eisenstein series \(E_k^{\chi,\psi}\), these values have a single trigonometric-sum expression involving \(\cot_{\widehat\chi}\) and \(\cot^{(k-2)}\). 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-\(2\) \(\mathrm{SU}(2)\)-TQFT case, the same trigonometric expressions coincide with the signature formula and imply its quantum modularity [2602.16159].

Source: https://www.emergentmind.com/topics/generalized-dedekind-sums