Cyclotomic Euler Sums Overview

• Cyclotomic Euler sums are convergent harmonic-type series defined using roots of unity, cyclotomic factorization, and evaluations of cyclotomic polylogarithms.
• They connect classical Euler sums with advanced constructs like Mellin transforms, shuffle relations, and analytic continuation to derive finite basis representations.
• These sums play a key role in arithmetic and p-adic studies, linking multiple zeta values with non-vanishing results and deep number-theoretic applications.

Searching arXiv for recent and foundational work on cyclotomic Euler sums. Cyclotomic Euler sums are convergent harmonic-type quantities controlled by roots of unity. In one standard usage, they are the $N\to\infty$ special constants obtained from cyclotomic harmonic sums and cyclotomic polylogarithms, in direct analogy with the way multiple zeta values arise from ordinary harmonic sums and harmonic polylogarithms (Ablinger et al., 2013). In a second, closely related usage, they are root-of-unity–twisted series of the form

$$S_{p_1,\ldots,p_k;q}(x_1,\ldots,x_k;x):=\sum_{n=1}^\infty \frac{\zeta_n(p_1;x_1)\cdots \zeta_n(p_k;x_k)}{n^q}\,x^n,$$

together with level-$2$, Hurwitz-type, and $t$- or $T$-analogous variants (Rui et al., 30 Aug 2025). Across these formulations, the subject is organized by four recurring mechanisms: cyclotomic factorization of $x^m-1$, Mellin and inverse Mellin correspondences between nested sums and iterated integrals, shuffle and quasi-shuffle reduction, and contour-integral parity identities.

1. Terminological scope and basic definitions

A persistent source of terminological variation is that different papers isolate different representatives of the same cyclotomic structure. In the survey treatment of Ablinger, Blümlein, and Schneider, “cyclotomic Euler sums” are the special constants

$$\sigma_{\cdots}=\lim_{N\to\infty}(\text{cyclotomic or generalized cyclotomic sum}),$$

and are encoded by cyclotomic polylogarithms evaluated at special points, typically $x=1$, where defined (Ablinger et al., 2013). In the contour-integral literature, by contrast, a generalized Euler sum becomes cyclotomic precisely when the parameters $x_1,\dots,x_k,x$ are roots of unity (Rui et al., 30 Aug 2025).

The broadest nested-sum definition used in the cyclotomic-harmonic framework is

$$S_{\{a_1,b_1,c_1\},\ldots,\{a_l,b_l,c_l\}}(s_1,\ldots,s_l;N) = \sum_{k_1=1}^{N} \frac{s_1^k}{(a_1k_1+b_1)^{c_1}} S_{\{a_2,b_2,c_2\};\ldots;\{a_l,b_l,c_l\}}(s_2,\ldots,s_l;k_1), \qquad S_\emptyset=1,$$

with $$S_{p_1,\ldots,p_k;q}(x_1,\ldots,x_k;x):=\sum_{n=1}^\infty \frac{\zeta_n(p_1;x_1)\cdots \zeta_n(p_k;x_k)}{n^q}\,x^n,$$0, $$S_{p_1,\ldots,p_k;q}(x_1,\ldots,x_k;x):=\sum_{n=1}^\infty \frac{\zeta_n(p_1;x_1)\cdots \zeta_n(p_k;x_k)}{n^q}\,x^n,$$1, $$S_{p_1,\ldots,p_k;q}(x_1,\ldots,x_k;x):=\sum_{n=1}^\infty \frac{\zeta_n(p_1;x_1)\cdots \zeta_n(p_k;x_k)}{n^q}\,x^n,$$2, and weight $$S_{p_1,\ldots,p_k;q}(x_1,\ldots,x_k;x):=\sum_{n=1}^\infty \frac{\zeta_n(p_1;x_1)\cdots \zeta_n(p_k;x_k)}{n^q}\,x^n,$$3 (Ablinger et al., 2011). A simpler single-index specialization is

$$S_{p_1,\ldots,p_k;q}(x_1,\ldots,x_k;x):=\sum_{n=1}^\infty \frac{\zeta_n(p_1;x_1)\cdots \zeta_n(p_k;x_k)}{n^q}\,x^n,$$4

described as a harmonic sum with periodic gaps (Ablinger et al., 2013).

The non-embedded contour-integral formulation starts from the finite twisted harmonic numbers

$$S_{p_1,\ldots,p_k;q}(x_1,\ldots,x_k;x):=\sum_{n=1}^\infty \frac{\zeta_n(p_1;x_1)\cdots \zeta_n(p_k;x_k)}{n^q}\,x^n,$$5

and builds

$$S_{p_1,\ldots,p_k;q}(x_1,\ldots,x_k;x):=\sum_{n=1}^\infty \frac{\zeta_n(p_1;x_1)\cdots \zeta_n(p_k;x_k)}{n^q}\,x^n,$$6

with weight $$S_{p_1,\ldots,p_k;q}(x_1,\ldots,x_k;x):=\sum_{n=1}^\infty \frac{\zeta_n(p_1;x_1)\cdots \zeta_n(p_k;x_k)}{n^q}\,x^n,$$7 and order $$S_{p_1,\ldots,p_k;q}(x_1,\ldots,x_k;x):=\sum_{n=1}^\infty \frac{\zeta_n(p_1;x_1)\cdots \zeta_n(p_k;x_k)}{n^q}\,x^n,$$8 (Rui et al., 30 Aug 2025). The level-$$S_{p_1,\ldots,p_k;q}(x_1,\ldots,x_k;x):=\sum_{n=1}^\infty \frac{\zeta_n(p_1;x_1)\cdots \zeta_n(p_k;x_k)}{n^q}\,x^n,$$9 analogue replaces $2$0 by $2$1: $2$2 and is called a cyclotomic Euler $2$3-sum (Wang et al., 8 Sep 2025).

This suggests that “cyclotomic Euler sums” is best treated as a family name rather than a single rigid definition. The common content is root-of-unity twisting, cyclotomic denominators, and reduction to polylogarithmic or zeta-like constants.

2. Cyclotomic factorization and the elementary origin of the sums

A direct elementary source of cyclotomic Euler sums is the factorization

$2$4

The procedure emphasized by Sofo associates convergent series and definite integrals to this factorization by removing the trivial root $2$5, inverting the remaining factor, expanding it as a geometric series or product of quadratic factors, integrating term-by-term on $2$6, and interpreting the resulting periodic coefficients in terms of Dirichlet characters (Boya et al., 2015).

The prototype $2$7 yields

$2$8

while the quartic factor $2$9 gives

$t$0

For $t$1,

$t$2

and

$t$3

with the coefficient pattern identified with the nontrivial Dirichlet character modulo $t$4 (Boya et al., 2015).

The same mechanism persists for higher $t$5. If $t$6, then the nontrivial roots of $t$7 come in conjugate pairs, producing quadratic factors

$t$8

For prime $t$9,

$T$0

and the resulting harmonic-type series acquire periodic arithmetic coefficients. For composite $T$1, additional decompositions such as

$T$2

produce mixtures of lower cyclotomic contributions; in particular,

$T$3

reflecting the coexistence of the $T$4 and $T$5 patterns (Boya et al., 2015).

In this elementary setting, cyclotomic Euler sums appear as Dirichlet-$T$6-type evaluations in disguise. The periodic residue-class structure generated by roots of unity already contains the essential arithmetic of the more elaborate nested theories.

3. Cyclotomic harmonic sums, polylogarithms, and Mellin correspondence

The systematic framework replaces the classical harmonic-polylogarithm alphabet $T$7 by all cyclotomic letters

$T$8

where $T$9 is the $x^m-1$0th cyclotomic polynomial (Ablinger et al., 2013). Iterating these letters yields cyclotomic polylogarithms; iterating the corresponding Mellin-space denominators yields cyclotomic harmonic sums. The basic structural statement is that nested sums in Mellin space correspond, via inverse Mellin transform, to iterated integrals in $x^m-1$1-space (Ablinger et al., 2013).

In the more explicit construction of Ablinger, Blümlein, and Schneider, the cyclotomic harmonic polylogarithms $x^m-1$2 are Poincaré-iterated integrals over the alphabet

$x^m-1$3

and the finite cyclotomic sums admit Mellin representations by these letters (Ablinger et al., 2011). The paper also performs analytic continuation of cyclotomic harmonic sums to complex values of $x^m-1$4 using analytic representations and derives basis representations for weight $x^m-1$5 sums up to cyclotomy $x^m-1$6 (Ablinger et al., 2011).

The constants obtained at $x^m-1$7 or $x^m-1$8 extend multiple zeta values. For cyclotomy $x^m-1$9, the constants can be expressed in terms of

$$\sigma_{\cdots}=\lim_{N\to\infty}(\text{cyclotomic or generalized cyclotomic sum}),$$0

with $$\sigma_{\cdots}=\lim_{N\to\infty}(\text{cyclotomic or generalized cyclotomic sum}),$$1 replacing $$\sigma_{\cdots}=\lim_{N\to\infty}(\text{cyclotomic or generalized cyclotomic sum}),$$2 as a more fundamental quantity (Ablinger et al., 2013). For higher cyclotomy, logarithms such as

$$\sigma_{\cdots}=\lim_{N\to\infty}(\text{cyclotomic or generalized cyclotomic sum}),$$3

and algebraic numbers occur, and for $$\sigma_{\cdots}=\lim_{N\to\infty}(\text{cyclotomic or generalized cyclotomic sum}),$$4 and weight $$\sigma_{\cdots}=\lim_{N\to\infty}(\text{cyclotomic or generalized cyclotomic sum}),$$5 the constants include values such as $$\sigma_{\cdots}=\lim_{N\to\infty}(\text{cyclotomic or generalized cyclotomic sum}),$$6, $$\sigma_{\cdots}=\lim_{N\to\infty}(\text{cyclotomic or generalized cyclotomic sum}),$$7, $$\sigma_{\cdots}=\lim_{N\to\infty}(\text{cyclotomic or generalized cyclotomic sum}),$$8, $$\sigma_{\cdots}=\lim_{N\to\infty}(\text{cyclotomic or generalized cyclotomic sum}),$$9, and $x=1$0; the constant $x=1$1 is Catalan’s constant (Ablinger et al., 2013).

A further extension consists of infinite generalized harmonic sums at roots of unity,

$x=1$2

which are the complex-root-of-unity versions of the same special-constant theory (Ablinger et al., 2013).

4. Algebraic reduction, analytic continuation, and independence

Cyclotomic Euler sums inherit the two basic Hopf-algebraic structures of the subject: shuffle relations on the iterated-integral side and quasi-shuffle, or stuffle, relations on the nested-sum side. Cyclotomic harmonic polylogarithms satisfy shuffle algebra relations, while cyclotomic harmonic sums satisfy quasi-shuffle relations and additional structural identities coming from differentiation, duplication, and multiple-argument relations (Ablinger et al., 2013).

In the cyclotomic-sum framework, these relations are effective enough to produce finite bases. The 2011 construction shows that finite cyclotomic sums are meromorphic functions of $x=1$3, with poles at non-positive integers, because their Mellin representations are factorial series; analytic continuation to complex $x=1$4 is performed using first-order difference equations together with asymptotic expansions (Ablinger et al., 2011). This analytic continuation is essential in both symbolic summation and asymptotic analysis.

A stronger structural result is obtained in the difference-ring treatment of Schneider. Starting from the alphabet

$x=1$5

with basic summands

$x=1$6

the paper constructs reduced difference rings for harmonic, alternating, and cyclotomic harmonic sums and proves that the remaining basis sums are algebraically independent in the reduced polynomial algebra (Ablinger et al., 2015). More significantly, the canonical map into the ring of sequences is injective: for $x=1$7, the reduced difference ring $x=1$8 is an $x=1$9-extension of $x_1,\dots,x_k,x$0, and

$x_1,\dots,x_k,x$1

is a difference ring embedding (Ablinger et al., 2015).

This implies that the sequences produced by the basis cyclotomic harmonic sums are algebraically independent over the rational sequences adjoined with the alternating sequence $x_1,\dots,x_k,x$2 (Ablinger et al., 2015). For cyclotomic Euler sums, this matters because the constants at infinity are not isolated accidents: they sit atop a formally rigid nested-sum calculus with no hidden algebraic relations beyond the known reductions.

5. Contour integration and parity phenomena

A major recent development is a contour-integral theory of parity for cyclotomic Euler sums. The central analytic kernel is the generalized digamma-like series

$x_1,\dots,x_k,x$3

together with the extended trigonometric function

$x_1,\dots,x_k,x$4

which interpolates the classical kernels

$x_1,\dots,x_k,x$5

(Rui et al., 30 Aug 2025). Residues of products of $x_1,\dots,x_k,x$6, derivatives of $x_1,\dots,x_k,x$7, and rational factors encode cyclotomic Euler sums and their reflected versions under inversion of the root-of-unity parameters.

For the non-embedded cyclotomic Euler sums $x_1,\dots,x_k,x$8, the general parity theorem states that

$x_1,\dots,x_k,x$9

reduces to a combination of sums of lower orders (Rui et al., 30 Aug 2025). The same paper gives explicit formulas in linear, quadratic, and cubic cases and derives corresponding parity statements for multiple polylogarithms.

The level-$$S_{\{a_1,b_1,c_1\},\ldots,\{a_l,b_l,c_l\}}(s_1,\ldots,s_l;N) = \sum_{k_1=1}^{N} \frac{s_1^k}{(a_1k_1+b_1)^{c_1}} S_{\{a_2,b_2,c_2\};\ldots;\{a_l,b_l,c_l\}}(s_2,\ldots,s_l;k_1), \qquad S_\emptyset=1,$$0 analogue behaves similarly. For cyclotomic Euler $$S_{\{a_1,b_1,c_1\},\ldots,\{a_l,b_l,c_l\}}(s_1,\ldots,s_l;N) = \sum_{k_1=1}^{N} \frac{s_1^k}{(a_1k_1+b_1)^{c_1}} S_{\{a_2,b_2,c_2\};\ldots;\{a_l,b_l,c_l\}}(s_2,\ldots,s_l;k_1), \qquad S_\emptyset=1,$$1-sums,

$$S_{\{a_1,b_1,c_1\},\ldots,\{a_l,b_l,c_l\}}(s_1,\ldots,s_l;N) = \sum_{k_1=1}^{N} \frac{s_1^k}{(a_1k_1+b_1)^{c_1}} S_{\{a_2,b_2,c_2\};\ldots;\{a_l,b_l,c_l\}}(s_2,\ldots,s_l;k_1), \qquad S_\emptyset=1,$$2

reduces to a combination of lower-order sums, and in the linear and quadratic cases the paper supplies explicit formulas (Wang et al., 8 Sep 2025). Through stuffle relations, these results induce parity reductions for cyclotomic multiple $$S_{\{a_1,b_1,c_1\},\ldots,\{a_l,b_l,c_l\}}(s_1,\ldots,s_l;N) = \sum_{k_1=1}^{N} \frac{s_1^k}{(a_1k_1+b_1)^{c_1}} S_{\{a_2,b_2,c_2\};\ldots;\{a_l,b_l,c_l\}}(s_2,\ldots,s_l;k_1), \qquad S_\emptyset=1,$$3-values and for cyclotomic multiple $$S_{\{a_1,b_1,c_1\},\ldots,\{a_l,b_l,c_l\}}(s_1,\ldots,s_l;N) = \sum_{k_1=1}^{N} \frac{s_1^k}{(a_1k_1+b_1)^{c_1}} S_{\{a_2,b_2,c_2\};\ldots;\{a_l,b_l,c_l\}}(s_2,\ldots,s_l;k_1), \qquad S_\emptyset=1,$$4-values.

A parallel residue theory treats the Hurwitz-type variants

$$S_{\{a_1,b_1,c_1\},\ldots,\{a_l,b_l,c_l\}}(s_1,\ldots,s_l;N) = \sum_{k_1=1}^{N} \frac{s_1^k}{(a_1k_1+b_1)^{c_1}} S_{\{a_2,b_2,c_2\};\ldots;\{a_l,b_l,c_l\}}(s_2,\ldots,s_l;k_1), \qquad S_\emptyset=1,$$5

For these two classes, explicit parity formulas are proved in the linear and quadratic cases, and the results imply explicit parity formulas for cyclotomic multiple $$S_{\{a_1,b_1,c_1\},\ldots,\{a_l,b_l,c_l\}}(s_1,\ldots,s_l;N) = \sum_{k_1=1}^{N} \frac{s_1^k}{(a_1k_1+b_1)^{c_1}} S_{\{a_2,b_2,c_2\};\ldots;\{a_l,b_l,c_l\}}(s_2,\ldots,s_l;k_1), \qquad S_\emptyset=1,$$6-values and $$S_{\{a_1,b_1,c_1\},\ldots,\{a_l,b_l,c_l\}}(s_1,\ldots,s_l;N) = \sum_{k_1=1}^{N} \frac{s_1^k}{(a_1k_1+b_1)^{c_1}} S_{\{a_2,b_2,c_2\};\ldots;\{a_l,b_l,c_l\}}(s_2,\ldots,s_l;k_1), \qquad S_\emptyset=1,$$7-values up to depth three (Xu, 22 Sep 2025). The Hurwitz-shifted extension

$$S_{\{a_1,b_1,c_1\},\ldots,\{a_l,b_l,c_l\}}(s_1,\ldots,s_l;N) = \sum_{k_1=1}^{N} \frac{s_1^k}{(a_1k_1+b_1)^{c_1}} S_{\{a_2,b_2,c_2\};\ldots;\{a_l,b_l,c_l\}}(s_2,\ldots,s_l;k_1), \qquad S_\emptyset=1,$$8

introduces a reflection

$$S_{\{a_1,b_1,c_1\},\ldots,\{a_l,b_l,c_l\}}(s_1,\ldots,s_l;N) = \sum_{k_1=1}^{N} \frac{s_1^k}{(a_1k_1+b_1)^{c_1}} S_{\{a_2,b_2,c_2\};\ldots;\{a_l,b_l,c_l\}}(s_2,\ldots,s_l;k_1), \qquad S_\emptyset=1,$$9

and again yields explicit linear and quadratic parity formulas together with arbitrary-depth reduction theorems and conjectures for multiple Hurwitz polylogarithms (Rui, 30 Dec 2025).

A plausible implication is that parity is not an accidental low-depth phenomenon but a structural consequence of the cyclotomic residue calculus itself. The recent literature consistently presents the contour method as a unifying replacement for the classical cotangent-kernel argument of Flajolet–Salvy.

6. Arithmetic aspects, non-vanishing, and broader significance

The arithmetic side of the subject appears most clearly in the study of cyclotomic multiple harmonic sums

$$S_{p_1,\ldots,p_k;q}(x_1,\ldots,x_k;x):=\sum_{n=1}^\infty \frac{\zeta_n(p_1;x_1)\cdots \zeta_n(p_k;x_k)}{n^q}\,x^n,$$00

which serve as finite algebraic models for $$S_{p_1,\ldots,p_k;q}(x_1,\ldots,x_k;x):=\sum_{n=1}^\infty \frac{\zeta_n(p_1;x_1)\cdots \zeta_n(p_k;x_k)}{n^q}\,x^n,$$01-adic cyclotomic multiple zeta values (Jarossay, 2017). Jarossay proves non-vanishing of certain such sums by three distinct mechanisms: $$S_{p_1,\ldots,p_k;q}(x_1,\ldots,x_k;x):=\sum_{n=1}^\infty \frac{\zeta_n(p_1;x_1)\cdots \zeta_n(p_k;x_k)}{n^q}\,x^n,$$02-adic dominance using primes in short intervals, field-degree estimates for the roots of unity, and complex absolute-value arguments when the last exponent $$S_{p_1,\ldots,p_k;q}(x_1,\ldots,x_k;x):=\sum_{n=1}^\infty \frac{\zeta_n(p_1;x_1)\cdots \zeta_n(p_k;x_k)}{n^q}\,x^n,$$03 is sufficiently large (Jarossay, 2017).

These finite non-vanishing results feed directly into $$S_{p_1,\ldots,p_k;q}(x_1,\ldots,x_k;x):=\sum_{n=1}^\infty \frac{\zeta_n(p_1;x_1)\cdots \zeta_n(p_k;x_k)}{n^q}\,x^n,$$04-adic period theory. Via an explicit expansion expressing cyclotomic multiple harmonic sums as infinite sums of products of $$S_{p_1,\ldots,p_k;q}(x_1,\ldots,x_k;x):=\sum_{n=1}^\infty \frac{\zeta_n(p_1;x_1)\cdots \zeta_n(p_k;x_k)}{n^q}\,x^n,$$05-adic cyclotomic multiple zeta values, non-vanishing of the harmonic sums implies non-vanishing of certain A-adjoint $$S_{p_1,\ldots,p_k;q}(x_1,\ldots,x_k;x):=\sum_{n=1}^\infty \frac{\zeta_n(p_1;x_1)\cdots \zeta_n(p_k;x_k)}{n^q}\,x^n,$$06-adic cyclotomic multiple zeta values (Jarossay, 2017). In the special case $$S_{p_1,\ldots,p_k;q}(x_1,\ldots,x_k;x):=\sum_{n=1}^\infty \frac{\zeta_n(p_1;x_1)\cdots \zeta_n(p_k;x_k)}{n^q}\,x^n,$$07, an alternating-series argument gives a transparent non-vanishing theorem when $$S_{p_1,\ldots,p_k;q}(x_1,\ldots,x_k;x):=\sum_{n=1}^\infty \frac{\zeta_n(p_1;x_1)\cdots \zeta_n(p_k;x_k)}{n^q}\,x^n,$$08 and the remaining roots are $$S_{p_1,\ldots,p_k;q}(x_1,\ldots,x_k;x):=\sum_{n=1}^\infty \frac{\zeta_n(p_1;x_1)\cdots \zeta_n(p_k;x_k)}{n^q}\,x^n,$$09, making the link with alternating Euler-sum behavior explicit (Jarossay, 2017).

Taken together, these developments place cyclotomic Euler sums at the intersection of several mature theories. From cyclotomic factorization they inherit periodic Dirichlet-character patterns; from Mellin-transform technology they inherit iterated-integral representations and explicit basis reduction; from difference rings they inherit algebraic and sequence-theoretic independence; from contour integration they inherit parity reduction and reflection formulas; and from arithmetic geometry they inherit non-vanishing questions tied to $$S_{p_1,\ldots,p_k;q}(x_1,\ldots,x_k;x):=\sum_{n=1}^\infty \frac{\zeta_n(p_1;x_1)\cdots \zeta_n(p_k;x_k)}{n^q}\,x^n,$$10-adic cyclotomic multiple zeta values. The literature therefore treats them not as isolated evaluations, but as a coherent cyclotomic extension of the Euler-sum and multiple-zeta-value framework (Boya et al., 2015).