---
title: Cyclotomic Euler-Type Sums
url: https://www.emergentmind.com/topics/cyclotomic-euler-type-sums
type: topic
---

# Cyclotomic Euler-Type Sums

Cyclotomic Euler-type sums are infinite series built from finite harmonic, polylogarithmic, or odd-harmonic partial sums, weighted by denominators on integer or half-integer lattices and twisted by roots of unity. In current usage, the subject includes classical cyclotomic Euler sums \(S\), odd-denominator \(T\)-sums, Hurwitz-type \(R\)- and \(\tilde S\)-sums, and level-two Dirichlet-type extensions involving odd harmonic numbers. These families interpolate among classical Euler sums, alternating Euler sums, Hoffman \(t\)-values, Kaneko–Tsumura \(T\)-values, colored multiple zeta values, and multiple polylogarithms at roots of unity, with contour integration and residue calculus providing the dominant analytic technique for explicit evaluation and parity reduction [2509.00638] [2509.06706] [2509.17468] [2009.11704].

## 1. Definitions and nomenclature

The recent literature does not use a single canonical normalization. Instead, several closely related families are studied in parallel, all controlled by the same cyclotomic principle: finite nested sums are twisted by roots of unity and combined with denominator patterns that reflect integer, half-integer, or odd-denominator lattices [2509.00638] [2509.06706] [2509.17468].

For integers \(p_1,\dots,p_k,q\ge 1\), roots of unity \(x_1,\dots,x_k,x\), and finite polylogarithmic or \(t\)-polylogarithmic sums
\[
\zeta_n(p;x):=\sum_{m=1}^n \frac{x^m}{m^p},
\qquad
t_n(p;x):=\sum_{m=1}^n \frac{x^m}{(m-\tfrac12)^p},
\]
the principal families are as follows.

| Family | Prototype definition | Basic feature |
|---|---|---|
| \(S\)-sums | \(\displaystyle S_{p_1,\dots,p_k;q}(x_1,\dots,x_k;x)=\sum_{n=1}^\infty \frac{\zeta_n(p_1;x_1)\cdots \zeta_n(p_k;x_k)}{n^q}x^n\) | Integer denominator |
| \(T\)-sums | \(\displaystyle T_{p_1,\dots,p_k;q}(x_1,\dots,x_k;x)=\sum_{n=1}^\infty \frac{t_n(p_1;x_1)\cdots t_n(p_k;x_k)}{(n-\tfrac12)^q}x^n\) | Half-integer denominator |
| \(R\)-sums | \(\displaystyle R_{p_1,\dots,p_k;q}(x_1,\dots,x_k;x)=\sum_{n=0}^\infty \frac{\zeta_n(p_1;x_1)\cdots \zeta_n(p_k;x_k)}{(n+\tfrac12)^q}x^n\) | Hurwitz-type half-integer shift |
| \(\tilde S\)-sums | \(\displaystyle \tilde S_{p_1,\dots,p_k;q}(x_1,\dots,x_k;x)=\sum_{n=1}^\infty \frac{t_n(p_1;x_1)\cdots t_n(p_k;x_k)}{n^q}x^n\) | Integer denominator with odd-harmonic numerators |

If \(x,x_1,\dots,x_k\) are \(N\)th roots of unity, these become cyclotomic Euler sums of level \(N\). The standard weight is
\[
w=p_1+\cdots+p_k+q,
\]
and the order, or Euler-sum depth, is \(k\) [2509.00638] [2509.06706]. This order should be distinguished from the depth of the associated multiple polylogarithms or multiple \(t\)-values, although the two are linked by stuffle decompositions.

A useful point of comparison is the older cyclotomic harmonic-sum notation
\[
S_{l,m,n}(N)=\sum_{k=0}^N \frac{(\operatorname{sign}(n))^k}{(lk+m)^{|n|}},
\]
together with the more general nested cyclotomic sums
\[
S_{\{a_1,b_1,c_1\};\dots;\{a_\ell,b_\ell,c_\ell\}}(s_1,\dots,s_\ell;N),
\]
which already encode Euler-type alternation through roots-of-unity or sign weights [1310.5645].

## 2. Level-two origin: odd denominators and Dirichlet-type extensions

A large part of the modern theory grew out of level-two constructions, where odd denominators and mod-\(2\) or mod-\(4\) character twists replace the ordinary harmonic numbers. In this setting one introduces the shifted odd harmonic numbers
\[
h_n^{(s)}:=\sum_{k=1}^n \frac{1}{(k-\tfrac12)^s},
\qquad
\bar h_n^{(s)}:=\sum_{k=1}^n \frac{(-1)^{k-1}}{(k-\tfrac12)^s},
\]
and also the strictly odd-denominator normalization
\[
\mathfrak h_n^{(s)}:=\sum_{k=1}^n \frac{1}{(2k-1)^s},
\qquad
h_n^{(s)}=2^s\mathfrak h_n^{(s)}.
\]
The corresponding Euler \(T\)- and \(S\)-sums are
\[
T(p_1,\dots,p_k;q):=\sum_{n=1}^\infty \frac{h_n^{(p_1)}\cdots h_n^{(p_k)}}{(n-\tfrac12)^q},
\qquad
S(p_1,\dots,p_k;q):=\sum_{n=1}^\infty \frac{h_n^{(p_1)}\cdots h_n^{(p_k)}}{n^q},
\]
together with alternating variants obtained by replacing some \(h_n^{(p)}\) by \(\bar h_n^{(p)}\) and by inserting factors \((-1)^{n-1}\) [2009.11704].

These sums are tied to two level-two multiple-value theories. Hoffman's multiple \(t\)-values are defined by
\[
t(s_1,\dots,s_d):=\sum_{n_1>\cdots>n_d\ge 1}\prod_{j=1}^d \frac{1}{(n_j-\tfrac12)^{s_j}},
\]
while Kaneko–Tsumura \(T\)-values are multiple zeta values of level two:
\[
T(s_1,\dots,s_d)
=
2^d\sum_{n_1>\cdots>n_d\ge 1}
\frac{1}{(2n_1-d)^{s_1}\cdots(2n_d-1)^{s_d}}.
\]
The paper "Two Variants of Euler Sums" proves that every Euler \(T\)-sum of weight \(w\) and degree \(k\) is a rational linear combination of Hoffman \(t\)-values of weight \(w\) and depth at most \(k+1\), and develops alternating Euler \(T\)-sums and a second Euler-type family \(\breve S\) connected directly to Kaneko–Tsumura \(T\)-values [1906.07654].

The paper "Dirichlet type extensions of Euler sums" pushes this level-two theory further. It studies alternating Euler \(T\)-sums and \(\S\)-sums, establishes explicit formulas for linear and quadratic cases by residue computations, derives parity theorems for Hoffman's double and triple \(t\)-values and Kaneko–Tsumura's double and triple \(T\)-values, and shows that linear \(T\)-sums and \(\S\)-sums are expressible in terms of colored multiple zeta values [2009.11704]. In this framework, odd denominators and alternating signs are naturally linked to Dirichlet beta phenomena and to characters mod \(2\) and mod \(4\).

## 3. Polylogarithmic and cyclotomic realization

Cyclotomic Euler-type sums are not isolated series; they are systematic avatars of multiple polylogarithms at roots of unity. For the basic \(S\)-family one has the depth-one stuffle identity
\[
S_{p;q}(x;y)=\Li_{p,q}(x,y)+\Li_{p+q}(xy),
\]
and depth-two formulas express \(S_{p_1,p_2;q}(x_1,x_2;x)\) as a sum of triple and double multiple polylogarithms. Consequently, parity identities for Euler sums translate directly into depth-drop relations for cyclotomic multiple polylogarithms [2509.00638].

For odd-denominator variants the same mechanism persists. Cyclotomic Euler \(T\)-sums satisfy stuffle-type relations with cyclotomic multiple \(t\)-values; in particular,
\[
T_{p;q}(x;y)=\ti_{p,q}(x,y)+\ti_{p+q}(xy),
\]
where
\[
\ti_{k_1,\ldots,k_r}(x_1,\ldots,x_r)
:=
\sum_{0<n_1<\cdots<n_r}
\frac{x_1^{n_1}\cdots x_r^{n_r}}{(n_1-\tfrac12)^{k_1}\cdots(n_r-\tfrac12)^{k_r}}.
\]
This “non-embedded” formulation is central in the recent parity theory of cyclotomic Euler \(T\)-sums [2509.06706].

A further structural layer is provided by colored multiple zeta values. In "Dirichlet type extensions of Euler sums," the level-\(4\) notation
\[
L_4(s_1,\dots,s_d;t_1,\dots,t_d)
=
\operatorname{Li}_{s_1,\dots,s_d}(i^{t_1},\dots,i^{t_d}),
\qquad i=e^{\pi i/2},
\]
is used to express all linear \(T\)- and \(S\)-sums explicitly as linear combinations of level-\(4\) colored multiple zeta values. Thus level two odd-denominator and alternating phenomena are absorbed into level four colored polylogarithms [2009.11704].

The broader cyclotomic formalism is supplied by cyclotomic harmonic polylogarithms. Their alphabet is
\[
\mathfrak A=\left\{\frac1x\right\}\cup
\left\{\frac{x^l}{\Phi_k(x)}\;\middle|\;k\in\mathbb N_+,\ 0\le l<\varphi(k)\right\},
\]
with \(\Phi_k(x)\) the \(k\)th cyclotomic polynomial. Mellin transforms of iterated integrals over this alphabet generate cyclotomic harmonic sums, and special values at \(x=1\) or in the limit \(N\to\infty\) are related to colored multiple zeta values. Basis representations for weights \(w=1,2\) were derived up to cyclotomy \(l=20\) [1105.6063]. The survey literature emphasizes that this language subsumes alternating Euler sums, cyclotomic sums with roots-of-unity weights, and the associated constants such as Dirichlet beta values, polylogarithms at rational arguments, and polygamma values [1310.5645].

## 4. Parity theorems and explicit reductions

The modern theory is organized around parity: signed combinations of cyclotomic Euler-type sums reduce to lower-order objects. For the classical cyclotomic \(S\)-family, Rui and Xu proved that for roots of unity \(x,x_1,\dots,x_r\),
\[
(-1)^r
S_{p_1,\dots,p_r;q}\!\Big(x_1,\dots,x_r;(x x_1\cdots x_r)^{-1}\Big)
+
(-1)^{p_1+\cdots+p_r+q}
S_{p_1,\dots,p_r;q}\!\Big(x_1^{-1},\dots,x_r^{-1};x x_1\cdots x_r\Big)
\]
is a linear combination of cyclotomic Euler sums of strictly lower order. The same paper gives explicit closed formulas in the linear and quadratic cases and derives corresponding depth-drop statements for multiple polylogarithms [2509.00638].

An analogous theorem holds for cyclotomic Euler \(T\)-sums. Wang and Xu show that
\[
x\,T_{p_1,\dots,p_r;q}\!\Big(x_1,\dots,x_r;(x x_1\cdots x_r)^{-1}\Big)
+
(-1)^{p_1+\cdots+p_r+q+r}
T_{p_1,\dots,p_r;q}\!\Big(x_1^{-1},\dots,x_r^{-1};x x_1\cdots x_r\Big)
\]
reduces to a combination of lower-order cyclotomic Euler \(T\)-sums and cyclotomic Euler sums. They also give explicit parity formulas for linear and quadratic \(T\)-sums and deduce corollaries for double and triple cyclotomic multiple \(t\)-values and for cyclotomic multiple \(T\)-values [2509.06706].

Xu’s two-family extension produces comparable arbitrary-order parity theorems for
\[
R_{p_1,\dots,p_r;q}(x_1,\dots,x_r;x)
\quad\text{and}\quad
\tilde S_{p_1,\dots,p_r;q}(x_1,\dots,x_r;x),
\]
with explicit linear and quadratic identities. These formulas imply explicit parity criteria for cyclotomic multiple \(S\)-values and \(T\)-values up to depth three [2509.17468].

At level two, the parity phenomenon specializes to concrete reduction theorems. "Dirichlet type extensions of Euler sums" proves that linear \(T\)- and \(S\)-sums reduce to zeta values, \(t\)-values, Dirichlet beta values, \(\ln 2\), and colored multiple zeta values; quadratic sums reduce to combinations of lower-depth Euler sums and single or double zeta/\(t\)-objects; and parity theorems follow for Hoffman's double and triple \(t\)-values and for Kaneko–Tsumura double and triple \(T\)-values [2009.11704]. The earlier paper "Two Variants of Euler Sums" proves, in particular, that if the weight \(p_1+\cdots+p_k+q\) and the degree \(k\) have the same parity, then \(T_{p_1\cdots p_k,q}\) lies in the span of lower-degree \(T\)-sums, \(\log(2)\), and multiple zeta values of bounded depth [1906.07654].

Representative explicit evaluations include
\[
S(1,2)=\frac12\zeta(3),
\qquad
T(1,2)=-\frac72\zeta(3)+\frac{\pi^2}{2}\ln 2,
\qquad
T(2,2)=24\,\zeta(4),
\]
and the odd-denominator identity
\[
\sum_{n\ge 1}\frac{(\mathfrak h_n^{(1)})^2}{(2n-1)^2}
=
\frac{\pi^2}{6}\ln^2 2+\frac{\pi^4}{6},
\]
which is rederived in the Dirichlet-type framework [2009.11704].

## 5. Contour integration and residue calculus

The dominant method throughout the subject is the Flajolet–Salvy residue paradigm, generalized to cyclotomic kernels. In the recent cyclotomic papers the key analytic objects are
\[
\phi(s;x):=\sum_{k=0}^\infty \frac{x^k}{k+s},
\qquad
\Phi(s;x):=\phi(s;x)-\phi(-s;x^{-1})-\frac1s.
\]
For roots of unity \(x\), every integer is a simple pole of \(\Phi(s;x)\); moreover,
\[
\Phi(s;1)=\pi\cot(\pi s),
\qquad
\Phi(s;-1)=\pi\csc(\pi s).
\]
These functions serve as kernel factors in contour integrals whose large-circle contribution vanishes [2509.00638] [2509.06706].

The residue identity has the form
\[
\sum_{\alpha\in O}\operatorname{Res}(r(s)\xi(s),\alpha)
+
\sum_{\beta\in S}\operatorname{Res}(r(s)\xi(s),\beta)
=
0,
\]
where \(\xi(s)\) is a meromorphic kernel and \(r(s)\) is a rational factor such as \(s^{-q}\), \((s+a)^{-q}\), or \((s+\tfrac12)^{-q}\). For cyclotomic Euler \(S\)-sums one integrates
\[
\oint_{(\infty)}
\Phi(s;x)\,
\prod_{j=1}^r
\frac{\phi^{(p_j-1)}(s;x_j)}{(p_j-1)!}\,
\frac{(-1)^{p_1+\cdots+p_r-r}}{s^q}\,ds=0,
\]
while for cyclotomic Euler \(T\)-sums the half-integer shift appears:
\[
\oint_{(\infty)}
\frac{\Phi(s;x)\,\phi^{(p_1-1)}(s+\tfrac12;x_1)\cdots\phi^{(p_r-1)}(s+\tfrac12;x_r)}
{(p_1-1)!\cdots(p_r-1)!\,(s+\tfrac12)^q}
(-1)^{p_1+\cdots+p_r-r}\,ds=0.
\]
Residues at integers, half-integers, and the extra poles of the rational factor generate the parity identities and explicit reduction formulas [2509.00638] [2509.06706] [2509.17468].

The level-two predecessor uses closely related kernels. "Two Variants of Euler Sums" employs \(\pi\tan(\pi s)\) for odd-denominator selection and \(\pi\cos(\pi s)\) for alternating variants, together with a shifted digamma-type function \(\Psi(1/2-s)\) whose local expansions encode odd harmonic numbers [1906.07654]. "Dirichlet type extensions of Euler sums" formulates the same strategy through parametric digamma and cotangent functions,
\[
\psi(-s;A)
\quad\text{and}\quad
\pi\cot(\pi s;A),
\]
with \(A_1=\{1\}\) and \(A_2=\{(-1)^k\}\), and base factors \(r(s)=s^{-q}\) or \(r(s)=(s-\tfrac12)^{-q}\) [2009.11704]. The common analytic pattern is therefore stable across classical, alternating, odd-denominator, cyclotomic, and Hurwitz-shifted settings.

## 6. Broader landscape, related structures, and open directions

Cyclotomic Euler-type sums sit inside a much larger cyclotomic-combinatorial framework. Cyclotomic harmonic sums and cyclotomic harmonic polylogarithms were developed partly because massive higher-order perturbative calculations require nested sums with denominators of the form \((\pm1)^k/(lk+m)^n\), and Mellin transforms of cyclotomic iterated integrals provide the natural analytic representation [1105.6063]. The survey literature extends this picture to generalized cyclotomic sums, roots-of-unity weighted sums, binomially weighted variants, and special constants such as \(\beta_l(1)\), \(\operatorname{Li}_k\) at rational arguments, and polygamma values at rational points [1310.5645].

A current extension is the Hurwitz-type theory. The paper "Contour Integrations and Parity Results of Hurwitz-type Cyclotomic Euler Sums" introduces shifted families
\[
S^{(a)}_{p_1,\dots,p_r;q},
\qquad
\tilde S^{(a)}_{p_1,\dots,p_r;q},
\qquad
R^{(a)}_{p_1,\dots,p_r;q},
\]
with \(a\notin\mathbb Z\), proves explicit linear and quadratic parity formulas, and formulates arbitrary-order parity theorems. Through their connection with multiple Hurwitz polylogarithms, these results lead to depth-drop statements and to conjectures asserting parity and symmetry for multiple Hurwitz polylogarithms of arbitrary depth [2601.00035].

A parallel, more elementary-looking but structurally related direction studies sums of the form
\[
\sum_{k=1}^\infty R(k)H_k,
\]
where denominators \((mk+n)^p\) act as residue-class filters modulo \(m\). In that setting, the evaluations reduce to digamma and polygamma values at rational points \(n/m\), hence to Hurwitz zeta values and Dirichlet \(L\)-values. This reinforces the cyclotomic viewpoint even outside the multiple-sum contour framework [2604.02384].

Several open problems remain explicit in the recent literature. "Dirichlet type extensions of Euler sums" asks whether certain linear \(R\)-sums \(R_{+1,\sigma}(1;q)\) with even \(q\) can be reduced entirely to combinations of classical zeta values, Dirichlet beta values, and double \(T\)-values [2009.11704]. "Contour Integrations and Parity Results of Cyclotomic Euler \(T\)-Sums and Multiple \(t\)-Values" proposes “mod products” parity conjectures for cyclotomic multiple \(t\)-values and cyclotomic multiple \(T\)-values, and asks whether only lower-depth cyclotomic multiple \(t\)-values, rather than general cyclotomic multiple zeta values, are needed in such reductions [2509.06706]. Xu’s paper on \(R\)- and \(\tilde S\)-sums formulates a stronger parity conjecture for cyclotomic multiple \(R\)-values beyond the weak parity presently deduced from Panzer-type polylogarithmic depth-drop [2509.17468].

The subject is therefore both structurally mature and actively expanding. Its core facts are now clear: cyclotomic Euler-type sums form a hierarchy of root-of-unity twisted Euler sums; they admit systematic contour-integral representations; parity forces reductions to lower order; and the resulting identities connect directly to multiple polylogarithms, multiple \(t\)- and \(T\)-values, colored multiple zeta values, and cyclotomic special constants.

Source: https://www.emergentmind.com/topics/cyclotomic-euler-type-sums