Cyclotomic Euler-Type Sums

• Cyclotomic Euler-type sums are infinite series built from finite nested harmonic and polylogarithmic sums twisted by roots of unity and weighted by integer or half-integer denominators.
• The topic encompasses families such as S-sums, T-sums, R-sums, and ~S-sums that interpolate between classical Euler sums, alternating sums, Hoffman t-values, and colored multiple zeta values.
• Analytic techniques like contour integration and residue calculus yield explicit evaluations and parity reductions, linking these sums to multiple polylogarithms and Hurwitz-type extensions.

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 (Rui et al., 30 Aug 2025, Wang et al., 8 Sep 2025, Xu, 22 Sep 2025, Xu et al., 2020).

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 (Rui et al., 30 Aug 2025, Wang et al., 8 Sep 2025, Xu, 22 Sep 2025).

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
$T$0-sums	$T$1	Integer denominator
$T$2-sums	$T$3	Half-integer denominator
$T$4-sums	$T$5	Hurwitz-type half-integer shift
$T$6-sums	$T$7	Integer denominator with odd-harmonic numerators

If $T$8 are $T$9th roots of unity, these become cyclotomic Euler sums of level $R$0. The standard weight is

$R$1

and the order, or Euler-sum depth, is $R$2 (Rui et al., 30 Aug 2025, Wang et al., 8 Sep 2025). This order should be distinguished from the depth of the associated multiple polylogarithms or multiple $R$3-values, although the two are linked by stuffle decompositions.

A useful point of comparison is the older cyclotomic harmonic-sum notation

$R$4

together with the more general nested cyclotomic sums

$R$5

which already encode Euler-type alternation through roots-of-unity or sign weights (Ablinger et al., 2013).

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-$R$6 or mod-$R$7 character twists replace the ordinary harmonic numbers. In this setting one introduces the shifted odd harmonic numbers

$R$8

and also the strictly odd-denominator normalization

$R$9

The corresponding Euler $\tilde S$0- and $\tilde S$1-sums are

$\tilde S$2

together with alternating variants obtained by replacing some $\tilde S$3 by $\tilde S$4 and by inserting factors $\tilde S$5 (Xu et al., 2020).

These sums are tied to two level-two multiple-value theories. Hoffman's multiple $\tilde S$6-values are defined by

$\tilde S$7

while Kaneko–Tsumura $\tilde S$8-values are multiple zeta values of level two: $\tilde S$9 The paper "Two Variants of Euler Sums" proves that every Euler $t$0-sum of weight $t$1 and degree $t$2 is a rational linear combination of Hoffman $t$3-values of weight $t$4 and depth at most $t$5, and develops alternating Euler $t$6-sums and a second Euler-type family $t$7 connected directly to Kaneko–Tsumura $t$8-values (Xu et al., 2019).

The paper "Dirichlet type extensions of Euler sums" pushes this level-two theory further. It studies alternating Euler $t$9-sums and $T$0-sums, establishes explicit formulas for linear and quadratic cases by residue computations, derives parity theorems for Hoffman's double and triple $T$1-values and Kaneko–Tsumura's double and triple $T$2-values, and shows that linear $T$3-sums and $T$4-sums are expressible in terms of colored multiple zeta values (Xu et al., 2020). In this framework, odd denominators and alternating signs are naturally linked to Dirichlet beta phenomena and to characters mod $T$5 and mod $T$6.

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 $T$7-family one has the depth-one stuffle identity

$T$8

and depth-two formulas express $T$9 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 (Rui et al., 30 Aug 2025).

For odd-denominator variants the same mechanism persists. Cyclotomic Euler $p_1,\dots,p_k,q\ge 1$0-sums satisfy stuffle-type relations with cyclotomic multiple $p_1,\dots,p_k,q\ge 1$1-values; in particular,

$p_1,\dots,p_k,q\ge 1$2

where

$p_1,\dots,p_k,q\ge 1$3

This “non-embedded” formulation is central in the recent parity theory of cyclotomic Euler $p_1,\dots,p_k,q\ge 1$4-sums (Wang et al., 8 Sep 2025).

A further structural layer is provided by colored multiple zeta values. In "Dirichlet type extensions of Euler sums," the level-$p_1,\dots,p_k,q\ge 1$5 notation

$p_1,\dots,p_k,q\ge 1$6

is used to express all linear $p_1,\dots,p_k,q\ge 1$7- and $p_1,\dots,p_k,q\ge 1$8-sums explicitly as linear combinations of level-$p_1,\dots,p_k,q\ge 1$9 colored multiple zeta values. Thus level two odd-denominator and alternating phenomena are absorbed into level four colored polylogarithms (Xu et al., 2020).

The broader cyclotomic formalism is supplied by cyclotomic harmonic polylogarithms. Their alphabet is

$x_1,\dots,x_k,x$0

with $x_1,\dots,x_k,x$1 the $x_1,\dots,x_k,x$2th cyclotomic polynomial. Mellin transforms of iterated integrals over this alphabet generate cyclotomic harmonic sums, and special values at $x_1,\dots,x_k,x$3 or in the limit $x_1,\dots,x_k,x$4 are related to colored multiple zeta values. Basis representations for weights $x_1,\dots,x_k,x$5 were derived up to cyclotomy $x_1,\dots,x_k,x$6 (Ablinger et al., 2011). 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 (Ablinger et al., 2013).

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 $x_1,\dots,x_k,x$7-family, Rui and Xu proved that for roots of unity $x_1,\dots,x_k,x$8,

$x_1,\dots,x_k,x$9

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 (Rui et al., 30 Aug 2025).

An analogous theorem holds for cyclotomic Euler $t$0-sums. Wang and Xu show that

$t$1

reduces to a combination of lower-order cyclotomic Euler $t$2-sums and cyclotomic Euler sums. They also give explicit parity formulas for linear and quadratic $t$3-sums and deduce corollaries for double and triple cyclotomic multiple $t$4-values and for cyclotomic multiple $t$5-values (Wang et al., 8 Sep 2025).

Xu’s two-family extension produces comparable arbitrary-order parity theorems for

$t$6

with explicit linear and quadratic identities. These formulas imply explicit parity criteria for cyclotomic multiple $t$7-values and $t$8-values up to depth three (Xu, 22 Sep 2025).

At level two, the parity phenomenon specializes to concrete reduction theorems. "Dirichlet type extensions of Euler sums" proves that linear $t$9- and $$\zeta_n(p;x):=\sum_{m=1}^n \frac{x^m}{m^p}, \qquad t_n(p;x):=\sum_{m=1}^n \frac{x^m}{(m-\tfrac12)^p},$$0-sums reduce to zeta values, $$\zeta_n(p;x):=\sum_{m=1}^n \frac{x^m}{m^p}, \qquad t_n(p;x):=\sum_{m=1}^n \frac{x^m}{(m-\tfrac12)^p},$$1-values, Dirichlet beta values, $$\zeta_n(p;x):=\sum_{m=1}^n \frac{x^m}{m^p}, \qquad t_n(p;x):=\sum_{m=1}^n \frac{x^m}{(m-\tfrac12)^p},$$2, and colored multiple zeta values; quadratic sums reduce to combinations of lower-depth Euler sums and single or double zeta/$$\zeta_n(p;x):=\sum_{m=1}^n \frac{x^m}{m^p}, \qquad t_n(p;x):=\sum_{m=1}^n \frac{x^m}{(m-\tfrac12)^p},$$3-objects; and parity theorems follow for Hoffman's double and triple $$\zeta_n(p;x):=\sum_{m=1}^n \frac{x^m}{m^p}, \qquad t_n(p;x):=\sum_{m=1}^n \frac{x^m}{(m-\tfrac12)^p},$$4-values and for Kaneko–Tsumura double and triple $$\zeta_n(p;x):=\sum_{m=1}^n \frac{x^m}{m^p}, \qquad t_n(p;x):=\sum_{m=1}^n \frac{x^m}{(m-\tfrac12)^p},$$5-values (Xu et al., 2020). The earlier paper "Two Variants of Euler Sums" proves, in particular, that if the weight $$\zeta_n(p;x):=\sum_{m=1}^n \frac{x^m}{m^p}, \qquad t_n(p;x):=\sum_{m=1}^n \frac{x^m}{(m-\tfrac12)^p},$$6 and the degree $$\zeta_n(p;x):=\sum_{m=1}^n \frac{x^m}{m^p}, \qquad t_n(p;x):=\sum_{m=1}^n \frac{x^m}{(m-\tfrac12)^p},$$7 have the same parity, then $$\zeta_n(p;x):=\sum_{m=1}^n \frac{x^m}{m^p}, \qquad t_n(p;x):=\sum_{m=1}^n \frac{x^m}{(m-\tfrac12)^p},$$8 lies in the span of lower-degree $$\zeta_n(p;x):=\sum_{m=1}^n \frac{x^m}{m^p}, \qquad t_n(p;x):=\sum_{m=1}^n \frac{x^m}{(m-\tfrac12)^p},$$9-sums, $T$00, and multiple zeta values of bounded depth (Xu et al., 2019).

Representative explicit evaluations include

$T$01

and the odd-denominator identity

$T$02

which is rederived in the Dirichlet-type framework (Xu et al., 2020).

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

$T$03

For roots of unity $T$04, every integer is a simple pole of $T$05; moreover,

$T$06

These functions serve as kernel factors in contour integrals whose large-circle contribution vanishes (Rui et al., 30 Aug 2025, Wang et al., 8 Sep 2025).

The residue identity has the form

$T$07

where $T$08 is a meromorphic kernel and $T$09 is a rational factor such as $T$10, $T$11, or $T$12. For cyclotomic Euler $T$13-sums one integrates

$T$14

while for cyclotomic Euler $T$15-sums the half-integer shift appears: $T$16 Residues at integers, half-integers, and the extra poles of the rational factor generate the parity identities and explicit reduction formulas (Rui et al., 30 Aug 2025, Wang et al., 8 Sep 2025, Xu, 22 Sep 2025).

The level-two predecessor uses closely related kernels. "Two Variants of Euler Sums" employs $T$17 for odd-denominator selection and $T$18 for alternating variants, together with a shifted digamma-type function $T$19 whose local expansions encode odd harmonic numbers (Xu et al., 2019). "Dirichlet type extensions of Euler sums" formulates the same strategy through parametric digamma and cotangent functions,

$T$20

with $T$21 and $T$22, and base factors $T$23 or $T$24 (Xu et al., 2020). 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 $T$25, and Mellin transforms of cyclotomic iterated integrals provide the natural analytic representation (Ablinger et al., 2011). The survey literature extends this picture to generalized cyclotomic sums, roots-of-unity weighted sums, binomially weighted variants, and special constants such as $T$26, $T$27 at rational arguments, and polygamma values at rational points (Ablinger et al., 2013).

A current extension is the Hurwitz-type theory. The paper "Contour Integrations and Parity Results of Hurwitz-type Cyclotomic Euler Sums" introduces shifted families

$T$28

with $T$29, 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 (Rui, 30 Dec 2025).

A parallel, more elementary-looking but structurally related direction studies sums of the form

$T$30

where denominators $T$31 act as residue-class filters modulo $T$32. In that setting, the evaluations reduce to digamma and polygamma values at rational points $T$33, hence to Hurwitz zeta values and Dirichlet $T$34-values. This reinforces the cyclotomic viewpoint even outside the multiple-sum contour framework (Bailey et al., 1 Apr 2026).

Several open problems remain explicit in the recent literature. "Dirichlet type extensions of Euler sums" asks whether certain linear $T$35-sums $T$36 with even $T$37 can be reduced entirely to combinations of classical zeta values, Dirichlet beta values, and double $T$38-values (Xu et al., 2020). "Contour Integrations and Parity Results of Cyclotomic Euler $T$39-Sums and Multiple $T$40-Values" proposes “mod products” parity conjectures for cyclotomic multiple $T$41-values and cyclotomic multiple $T$42-values, and asks whether only lower-depth cyclotomic multiple $T$43-values, rather than general cyclotomic multiple zeta values, are needed in such reductions (Wang et al., 8 Sep 2025). Xu’s paper on $T$44- and $T$45-sums formulates a stronger parity conjecture for cyclotomic multiple $T$46-values beyond the weak parity presently deduced from Panzer-type polylogarithmic depth-drop (Xu, 22 Sep 2025).

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$47- and $T$48-values, colored multiple zeta values, and cyclotomic special constants.