Cyclotomic Multiple Harmonic Values

• CMHVs are nested harmonic sums twisted by roots of unity, defined by weight and depth, and serve as a unifying framework across various number theoretic contexts.
• They provide explicit coordinates for p-adic cyclotomic multiple zeta values through prime-weighted families and exhibit rich algebraic structures via shuffle and quasi-shuffle relations.
• CMHVs connect finite q-series, polylogarithmic special values, and multiple L-function theories, facilitating computational methods in fields like quantum field theory.

Cyclotomic multiple harmonic values (CMHVs) are nested harmonic objects twisted by roots of unity and organized by weight and depth. In the $p$-adic literature, a standard form is the family

$$\operatorname{MHV}\big((n_i)_r;(\epsilon_i)_r\big) = \left( p^{n_1+\cdots+n_r} H_p\big((n_i)_r;(\epsilon_i)_r\big) \right)_{p\in \mathcal P_c} \in \prod_{p\in\mathcal P_c}\mathbb Q_p(\mu_c),$$

where $H_p$ is a cyclotomic multiple harmonic sum and $\mathcal P_c$ is the set of primes not dividing $c$ (Furusho et al., 2018). In adjacent literatures, the same expression also denotes cyclotomic analogues of finite multiple zeta values built from finite multiple harmonic $q$-series at roots of unity, or special values of cyclotomic harmonic sums and polylogarithms at $N\to\infty$ or $x=1$ [(Bachmann et al., 2017); (Ablinger et al., 2011)]. Across these usages, the common structure is a root-of-unity twist, nested summation, and algebraic control by shuffle or quasi-shuffle formalisms.

1. Terminology, normalizations, and basic definitions

The term “CMHV” is not attached to a single normalization across the literature. Three recurrent frameworks are especially prominent.

Framework	Representative object	Role
$p$-adic/adelic	$\left(p^{\mathrm{wt}}H_p(\cdots)\right)_p$	Prime-weighted family in $$\operatorname{MHV}\big((n_i)_r;(\epsilon_i)_r\big) = \left( p^{n_1+\cdots+n_r} H_p\big((n_i)_r;(\epsilon_i)_r\big) \right)_{p\in \mathcal P_c} \in \prod_{p\in\mathcal P_c}\mathbb Q_p(\mu_c),$$0
Finite $$\operatorname{MHV}\big((n_i)_r;(\epsilon_i)_r\big) = \left( p^{n_1+\cdots+n_r} H_p\big((n_i)_r;(\epsilon_i)_r\big) \right)_{p\in \mathcal P_c} \in \prod_{p\in\mathcal P_c}\mathbb Q_p(\mu_c),$$1-series/cyclotomic FMZV	$$\operatorname{MHV}\big((n_i)_r;(\epsilon_i)_r\big) = \left( p^{n_1+\cdots+n_r} H_p\big((n_i)_r;(\epsilon_i)_r\big) \right)_{p\in \mathcal P_c} \in \prod_{p\in\mathcal P_c}\mathbb Q_p(\mu_c),$$2	Cyclotomic analogue of FMZVs
Polylogarithmic/special-value	$$\operatorname{MHV}\big((n_i)_r;(\epsilon_i)_r\big) = \left( p^{n_1+\cdots+n_r} H_p\big((n_i)_r;(\epsilon_i)_r\big) \right)_{p\in \mathcal P_c} \in \prod_{p\in\mathcal P_c}\mathbb Q_p(\mu_c),$$3, or $$\operatorname{MHV}\big((n_i)_r;(\epsilon_i)_r\big) = \left( p^{n_1+\cdots+n_r} H_p\big((n_i)_r;(\epsilon_i)_r\big) \right)_{p\in \mathcal P_c} \in \prod_{p\in\mathcal P_c}\mathbb Q_p(\mu_c),$$4	Constants from cyclotomic sums and iterated integrals

For positive integers $$\operatorname{MHV}\big((n_i)_r;(\epsilon_i)_r\big) = \left( p^{n_1+\cdots+n_r} H_p\big((n_i)_r;(\epsilon_i)_r\big) \right)_{p\in \mathcal P_c} \in \prod_{p\in\mathcal P_c}\mathbb Q_p(\mu_c),$$5, roots of unity $$\operatorname{MHV}\big((n_i)_r;(\epsilon_i)_r\big) = \left( p^{n_1+\cdots+n_r} H_p\big((n_i)_r;(\epsilon_i)_r\big) \right)_{p\in \mathcal P_c} \in \prod_{p\in\mathcal P_c}\mathbb Q_p(\mu_c),$$6, and $$\operatorname{MHV}\big((n_i)_r;(\epsilon_i)_r\big) = \left( p^{n_1+\cdots+n_r} H_p\big((n_i)_r;(\epsilon_i)_r\big) \right)_{p\in \mathcal P_c} \in \prod_{p\in\mathcal P_c}\mathbb Q_p(\mu_c),$$7, the cyclotomic multiple harmonic sum is

$$\operatorname{MHV}\big((n_i)_r;(\epsilon_i)_r\big) = \left( p^{n_1+\cdots+n_r} H_p\big((n_i)_r;(\epsilon_i)_r\big) \right)_{p\in \mathcal P_c} \in \prod_{p\in\mathcal P_c}\mathbb Q_p(\mu_c),$$8

Here $$\operatorname{MHV}\big((n_i)_r;(\epsilon_i)_r\big) = \left( p^{n_1+\cdots+n_r} H_p\big((n_i)_r;(\epsilon_i)_r\big) \right)_{p\in \mathcal P_c} \in \prod_{p\in\mathcal P_c}\mathbb Q_p(\mu_c),$$9 is the depth and $H_p$0 is the weight (Furusho et al., 2018, Fan, 5 Jul 2026). In the prime-weighted $H_p$1-adic normalization, one forms the family $H_p$2, and each component has $H_p$3-adic valuation at least $H_p$4 (Furusho et al., 2018).

A closely related normalization, used extensively by Jarossay, builds the weight factor directly into the sum: $H_p$5 This version is also the coefficient extractor for cyclotomic multiple polylogarithms: $H_p$6 Accordingly, CMHVs can be viewed either as prime-weighted families of truncations or as adelic packages of weighted coefficients of cyclotomic multiple polylogarithms (Jarossay, 2014).

2. CMHVs in $H_p$7-adic multiple zeta and $H_p$8-function theory

A central role of CMHVs is their function as explicit coordinates for $H_p$9-adic cyclotomic multiple zeta values. Explicit formulas express $\mathcal P_c$0-adic cyclotomic multiple zeta values in terms of prime weighted cyclotomic multiple harmonic sums, and conversely; the bridge is supplied by pro-unipotent harmonic actions and comparison maps between “series” and “integral” realizations (Jarossay, 2015). Jarossay later formulated the dependence on Frobenius iteration through fixed-point and iteration equations such as

$\mathcal P_c$1

and

$\mathcal P_c$2

placing CMHVs inside a dynamical formalism for crystalline Frobenius on the pro-unipotent fundamental groupoid of $\mathcal P_c$3 (Jarossay, 2016).

The most explicit $\mathcal P_c$4-function application is the Furusho–Jarossay theorem on $\mathcal P_c$5-adic multiple $\mathcal P_c$6-functions. For tuples of positive integers $\mathcal P_c$7, the family

$\mathcal P_c$8

can be expressed as an infinite sum of $\mathcal P_c$9-linear combinations of CMHVs, with depth $c$0, weight tending to infinity, and uniform convergence in $c$1 (Furusho et al., 2018). For $c$2, this recovers a formula parallel to Washington’s formula for values of the Kubota–Leopoldt $c$3-adic $c$4-function (Furusho et al., 2018).

A further refinement identifies the coefficient system in those expansions. Furusho and Jarossay had expressed the coefficients through cyclotomic multiple Bernoulli numbers; an explicit 2026 reformulation rewrites those coefficients in terms of twisted multiple Bernoulli numbers, equivalently special values of generalized Euler–Zagier–Lerch type complex multiple zeta functions at tuples of non-positive integers (Fan, 5 Jul 2026). In this form, special values of $c$5-adic multiple $c$6-functions become infinite CMHV expansions with analytically recognizable coefficient data.

3. Algebraic structures and internal relations

CMHVs satisfy harmonic analogues of the standard double shuffle package. They were introduced together with adjoint cyclotomic multiple zeta values as two closely related variants of cyclotomic multiple zeta values, and were shown to satisfy harmonic quasi-shuffle and shuffle relations, as well as families of relations related to associators and Kashiwara–Vergne theory (Jarossay, 2014). In that framework, the ind-scheme defined by harmonic double shuffle equations forms a torsor under a pro-unipotent harmonic action (Jarossay, 2014).

Jarossay’s explicit theory of the crystalline pro-unipotent fundamental groupoid makes these relations calculable rather than merely formal. Harmonic Ihara actions and comparison maps are shown to be compatible with algebraic relations, and two “harmonic” versions of Besser–Furusho–Jafari’s theorem transfer quasi-shuffle relations of weighted multiple harmonic sums to adjoint quasi-shuffle relations of cyclotomic $c$7-adic multiple zeta values (Jarossay, 2016). This makes CMHVs a computationally explicit intermediary between nested sums and $c$8-adic periods.

In the finite $c$9-series realization, the star-versions satisfy a duality formula: $q$0 where $q$1 is the reverse of the Hoffman dual of $q$2 (Bachmann et al., 2017). The same setting carries deformed stuffle structures $q$3 and $q$4, and the resulting relations descend to finite and symmetric multiple zeta values (Bachmann et al., 2017).

A distinct, but structurally complementary, result comes from difference-ring theory. Cyclotomic harmonic sums can be reduced by the quasi-shuffle algebra to basis sums, embedded into a tower of $q$5-extensions with unchanged constants, and then faithfully embedded into the ring of sequences; the image sequences of the basis sums are algebraically independent over the rational sequences adjoined with the alternating sequence $q$6 (Ablinger et al., 2015). This yields the statement that all algebraic relations among these sequences are already generated by the quasi-shuffle algebra (Ablinger et al., 2015).

4. Finite, symmetric, and colored cyclotomic variants

One influential finite realization begins with finite multiple harmonic $q$7-series at roots of unity. For an index $q$8, one defines

$q$9

together with the star version $N\to\infty$0 (Bachmann et al., 2017). At a primitive $N\to\infty$1th root of unity $N\to\infty$2, these lie in $N\to\infty$3. The cyclotomic analogues of finite multiple zeta values are then

$N\to\infty$4

where $N\to\infty$5 is the cyclotomic adelic quotient ring introduced in that work (Bachmann et al., 2017).

These values interpolate between finite and symmetric multiple zeta phenomena. Theorem A states that reduction modulo the prime ideal above $N\to\infty$6 yields the finite multiple zeta value $N\to\infty$7, while the analytic limit $N\to\infty$8, $N\to\infty$9, yields numbers whose real parts agree with symmetric multiple zeta values modulo $x=1$0 (Bachmann et al., 2017). The same paper formulates a cyclotomic reformulation of the Kaneko–Zagier conjecture: there should exist a homomorphism from the cyclotomic space to $x=1$1 with the same kernel as the reduction map to finite multiple zeta values (Bachmann et al., 2017).

Tasaka extended this perspective to colored multiple zeta values of level $x=1$2. Truncated multiple harmonic $x=1$3-series of level $x=1$4 yield analytic limits $x=1$5 and algebraic limits $x=1$6, the symmetric and finite colored multiple zeta values of level $x=1$7, and both form commutative algebras over $x=1$8 satisfying reversal, harmonic product, and shuffle relations (Tasaka, 2019). The higher-level Kaneko–Zagier conjecture in this setting asserts that the natural map from symmetric to finite colored multiple zeta values has kernel generated by $x=1$9 (Tasaka, 2019).

Numerical evidence in the cyclotomic FMZV framework suggests stabilization of the weight-$p$0 dimensions for large primes. For $p$1, the reported upper bounds are

$p$2

supporting the expectation that cyclotomic relations are governed by the same double-shuffle and duality patterns that appear for FMZVs and SMZVs (Bachmann et al., 2017).

5. Polylogarithmic realizations, analytic methods, and applications

In the analytic-combinatorial literature, CMHVs appear as special constants attached to cyclotomic harmonic sums and cyclotomic harmonic polylogarithms generated by cyclotomic polynomials. The relevant alphabet is

$p$3

and the associated iterated integrals $p$4 generalize harmonic polylogarithms (Ablinger et al., 2011). In this language, CMHVs are special values at $p$5 or the $p$6 limits of cyclotomic harmonic sums,

$p$7

with algebraic structure supplied by stuffle, shuffle, multiple-argument, duplication, and distribution relations [(Ablinger et al., 2011); (Ablinger et al., 2013)].

This framework was developed partly for high-order perturbative quantum field theory. Cyclotomic harmonic sums and polylogarithms arise via Mellin and inverse Mellin transforms in the analysis of massive Feynman integrals, and basis representations were derived for weight $p$8 sums up to cyclotomy $p$9 (Ablinger et al., 2011). The associated constants extend classical MZV-type bases by cyclotomic quantities such as logarithms at algebraic arguments, values of polygamma functions at rational arguments, and Catalan’s constant [(Ablinger et al., 2011); (Ablinger et al., 2013)]. The survey literature emphasizes that, beyond the displayed structural relations, it is not proven at present whether these are all relations in the cyclotomic setting (Ablinger et al., 2013).

Cyclotomic multiple zeta and harmonic structures also appear in explicit evaluation problems. Sun’s binomial-harmonic series and analogues can be represented as cyclotomic multiple zeta values of levels $\left(p^{\mathrm{wt}}H_p(\cdots)\right)_p$0, i.e. as Goncharov multiple polylogarithms at roots of unity; the paper introduces auxiliary spaces $\left(p^{\mathrm{wt}}H_p(\cdots)\right)_p$1 to organize the corresponding polylogarithmic values (Zhou, 2023). A related 2025 line of work studies cyclotomic multiple $\left(p^{\mathrm{wt}}H_p(\cdots)\right)_p$2-values and $\left(p^{\mathrm{wt}}H_p(\cdots)\right)_p$3-values—defined as even-odd variants of cyclotomic multiple zeta values and explicitly interpreted as CMHVs—and proves parity statements reducing the parity difference to lower-depth values, with explicit formulas available up to depth three through contour integration and residue calculus (Xu, 22 Sep 2025).

6. Arithmetic properties, non-vanishing, and open directions

The $\left(p^{\mathrm{wt}}H_p(\cdots)\right)_p$4-adic normalization of CMHVs carries strong valuation control: for every prime $\left(p^{\mathrm{wt}}H_p(\cdots)\right)_p$5, the quantity $\left(p^{\mathrm{wt}}H_p(\cdots)\right)_p$6 has $\left(p^{\mathrm{wt}}H_p(\cdots)\right)_p$7-adic valuation at least $\left(p^{\mathrm{wt}}H_p(\cdots)\right)_p$8 (Furusho et al., 2018). Non-vanishing is also accessible. Jarossay proved non-vanishing criteria for certain cyclotomic multiple harmonic sums under three types of hypotheses: sufficiently large $\left(p^{\mathrm{wt}}H_p(\cdots)\right)_p$9 relative to the depth, sufficiently large and pairwise suitably independent degrees of cyclotomic fields generated by the roots of unity, or sufficiently large last weight $$\operatorname{MHV}\big((n_i)_r;(\epsilon_i)_r\big) = \left( p^{n_1+\cdots+n_r} H_p\big((n_i)_r;(\epsilon_i)_r\big) \right)_{p\in \mathcal P_c} \in \prod_{p\in\mathcal P_c}\mathbb Q_p(\mu_c),$$00 (Jarossay, 2017). Through an explicit formula expressing $$\operatorname{MHV}\big((n_i)_r;(\epsilon_i)_r\big) = \left( p^{n_1+\cdots+n_r} H_p\big((n_i)_r;(\epsilon_i)_r\big) \right)_{p\in \mathcal P_c} \in \prod_{p\in\mathcal P_c}\mathbb Q_p(\mu_c),$$01 as an infinite sum of products of $$\operatorname{MHV}\big((n_i)_r;(\epsilon_i)_r\big) = \left( p^{n_1+\cdots+n_r} H_p\big((n_i)_r;(\epsilon_i)_r\big) \right)_{p\in \mathcal P_c} \in \prod_{p\in\mathcal P_c}\mathbb Q_p(\mu_c),$$02-adic cyclotomic multiple zeta values, these non-vanishing results imply the non-vanishing of certain $$\operatorname{MHV}\big((n_i)_r;(\epsilon_i)_r\big) = \left( p^{n_1+\cdots+n_r} H_p\big((n_i)_r;(\epsilon_i)_r\big) \right)_{p\in \mathcal P_c} \in \prod_{p\in\mathcal P_c}\mathbb Q_p(\mu_c),$$03-adic cyclotomic multiple zeta values (Jarossay, 2017).

A persistent point of clarification is terminological rather than mathematical: the literature uses “CMHV” for several tightly connected, but not identical, constructions. One usage emphasizes prime-weighted $$\operatorname{MHV}\big((n_i)_r;(\epsilon_i)_r\big) = \left( p^{n_1+\cdots+n_r} H_p\big((n_i)_r;(\epsilon_i)_r\big) \right)_{p\in \mathcal P_c} \in \prod_{p\in\mathcal P_c}\mathbb Q_p(\mu_c),$$04-adic families of truncations; another emphasizes finite cyclotomic $$\operatorname{MHV}\big((n_i)_r;(\epsilon_i)_r\big) = \left( p^{n_1+\cdots+n_r} H_p\big((n_i)_r;(\epsilon_i)_r\big) \right)_{p\in \mathcal P_c} \in \prod_{p\in\mathcal P_c}\mathbb Q_p(\mu_c),$$05-series modulo primes; a third emphasizes special constants arising from cyclotomic harmonic sums and iterated integrals [(Furusho et al., 2018); (Bachmann et al., 2017); (Ablinger et al., 2011)]. The overlap is substantial—each setting centers on nested cyclotomic harmonic data—but the surrounding coefficient rings, comparison maps, and conjectural relation spaces differ.

Several structural questions remain open. In the cyclotomic FMZV/SMZV direction, the conjectural equality of kernels between the finite and symmetric projections is still a central organizing principle (Bachmann et al., 2017). In the colored higher-level setting, the corresponding conjecture asserts that the kernel of the map from symmetric to finite colored multiple zeta values is generated by $$\operatorname{MHV}\big((n_i)_r;(\epsilon_i)_r\big) = \left( p^{n_1+\cdots+n_r} H_p\big((n_i)_r;(\epsilon_i)_r\big) \right)_{p\in \mathcal P_c} \in \prod_{p\in\mathcal P_c}\mathbb Q_p(\mu_c),$$06 (Tasaka, 2019). In the polylogarithmic special-value setting, the full set of relations among cyclotomic constants is not known (Ablinger et al., 2013). This suggests that CMHVs are best viewed not as a single isolated invariant, but as a common cyclotomic-harmonic interface linking finite, complex, and $$\operatorname{MHV}\big((n_i)_r;(\epsilon_i)_r\big) = \left( p^{n_1+\cdots+n_r} H_p\big((n_i)_r;(\epsilon_i)_r\big) \right)_{p\in \mathcal P_c} \in \prod_{p\in\mathcal P_c}\mathbb Q_p(\mu_c),$$07-adic multiple zeta and $$\operatorname{MHV}\big((n_i)_r;(\epsilon_i)_r\big) = \left( p^{n_1+\cdots+n_r} H_p\big((n_i)_r;(\epsilon_i)_r\big) \right)_{p\in \mathcal P_c} \in \prod_{p\in\mathcal P_c}\mathbb Q_p(\mu_c),$$08-value theories.