Cyclotomic Multiple Bernoulli Numbers (CMBNs)

• Cyclotomic Multiple Bernoulli Numbers (CMBNs) are defined as the coefficients in the expansion of cyclotomic multiple harmonic sums, serving as a bridge between p-adic multiple L-functions and complex multiple zeta functions.
• The explicit formula expresses CMBNs as a finite signed sum of twisted multiple Bernoulli numbers, which are themselves special values of generalized Euler-Zagier-Lerch zeta functions at non-positive integers.
• CMBNs play a critical role in the Furusho–Jarossay expansion by converting combinatorial data into explicit arithmetic coefficients essential for evaluating p-adic multiple L-function values.

Searching arXiv for the cited paper and closely related work on $p$-adic multiple $L$-functions, CMHVs, and multiple Bernoulli numbers. Search results indicate the primary directly relevant source is "p-adic multiple L-functions and twisted multiple Bernoulli numbers" (Fan, 5 Jul 2026). I’ll ground the article in that paper and restrict concrete claims to the provided data. Cyclotomic Multiple Bernoulli Numbers (CMBNs) are coefficients attached to cyclotomic multiple harmonic sums, and in the setting of $p$-adic multiple $L$-functions they serve as the coefficient system linking cyclotomic multiple harmonic values (CMHVs) to special values of $p$-adic multiple $L$-functions. In " $p$-adic multiple $L$-functions and twisted multiple Bernoulli numbers" (Fan, 5 Jul 2026), CMBNs are placed in a framework that also includes twisted multiple Bernoulli numbers (TMBNs), generalized Euler-Zagier-Lerch type complex multiple zeta functions, and the Furusho–Jarossay expansions of $p$-adic multiple $L$-function values. The central result is an explicit formula expressing CMBNs in terms of TMBNs, with the consequence that the relevant $L$0-adic multiple $L$1-function values can be written as infinite sums of CMHVs whose coefficients are special values of complex multiple zeta functions at tuples of non-positive integers.

1. Basic objects and notation

The paper organizes the subject around five families of quantities: twisted Bernoulli numbers, twisted multiple Bernoulli numbers, cyclotomic multiple Bernoulli numbers, cyclotomic multiple harmonic values, and special values of $L$2-adic multiple $L$3-functions (Fan, 5 Jul 2026).

For a root of unity $L$4, the twisted Bernoulli numbers $L$5 are defined by the generating function

$L$6

with the convention $L$7. This is the depth-$L$8 case.

For depth $L$9, let $p$0 and $p$1. The generating series

$p$2

is expanded as

$p$3

The coefficients $p$4 are the twisted multiple Bernoulli numbers.

For $p$5, $p$6, and $p$7, the cyclotomic multiple Bernoulli number

$p$8

is defined by the expansion

$p$9

where $L$0 is a cyclotomic multiple harmonic sum with certain combinatorial restrictions.

The CMHVs are built from the finite sums

$L$1

and the associated family

$L$2

The $L$3-adic multiple $L$4-function values studied in the paper are

$L$5

and the main $L$6-adic family is

$L$7

Object	Definition or role
$L$8	Twisted Bernoulli numbers defined by $L$9
$p$0	TMBNs from the multivariate generating function $p$1
$p$2	CMBNs from expansions of cyclotomic multiple harmonic sums
$p$3	CMHVs, families over $p$4
$p$5	$p$6-adic multiple $p$7-functions whose special values are expanded in the CMHV basis

2. Twisted multiple Bernoulli numbers and complex multiple zeta values

A basic structural point is that TMBNs are not introduced as isolated combinatorial coefficients; they are identified with special values of generalized Euler-Zagier-Lerch type complex multiple zeta functions at tuples of non-positive integers (Fan, 5 Jul 2026).

The paper uses the function

$p$8

For $p$9 and suitable $L$0, the cited theorem gives

$L$1

This relation is the mechanism by which TMBNs can be interpreted as special values of generalized Euler-Zagier-Lerch multiple zeta functions at tuples of non-positive integers. In the later expansion formulas, this allows the coefficient system to be rewritten from Bernoulli-type quantities to zeta-type quantities without changing the underlying combinatorics.

A plausible implication is that CMBNs inherit arithmetic significance not merely from their defining generating expansions, but from the fact that the TMBNs appearing in their explicit formula are already identified with special values of explicit complex multiple zeta functions.

3. Definition and combinatorial profile of CMBNs

The general definition of CMBNs is combinatorially involved. The paper formulates them through the coefficients in the expansion of the quantities

$L$2

where the data consist of indices $L$3, roots of unity $L$4, and step-shifts $L$5 (Fan, 5 Jul 2026).

The defining expansion is

$L$6

In this sense, CMBNs are coefficient objects attached to cyclotomic multiple harmonic sums rather than coefficients introduced by an independent generating function.

This placement is important for the later $L$7-adic theory. The coefficients that appear in the Furusho–Jarossay expansion of the $L$8-adic multiple $L$9-function values are precisely CMBNs of the form

$p$0

Thus the $p$1 stratum is the coefficient regime directly relevant to the special values under study.

A plausible misconception is to regard CMBNs as synonymous with TMBNs. The paper distinguishes them sharply: TMBNs arise as coefficients of the multivariate generating function $p$2, whereas CMBNs arise as coefficients in expansions of cyclotomic multiple harmonic sums. The main theorem then identifies the latter explicitly in terms of the former.

4. Explicit formula expressing CMBNs in terms of TMBNs

The main explicit formula, stated as Theorem 2.8, writes

$p$3

as a finite signed sum indexed by $p$4, auxiliary integers $p$5, and $p$6, with multinomial coefficients, powers of $p$7, Kronecker delta conditions on $p$8 and $p$9, and a TMBN term of the form

$L$0

(Fan, 5 Jul 2026).

The notation $L$1 is part of the combinatorial architecture: for $L$2 one uses a comma, and for $L$3 one uses addition, so the indexing system interpolates between keeping coordinates separate and merging them. The convention $L$4 is also used.

The coefficient relevant to the $L$5-adic applications is the special case $L$6, stated as Corollary 2.10: $L$7

In the wording of the paper’s summary, every CMBN of the type required for special values of $L$8-adic multiple $L$9-functions can be written explicitly as a signed sum of TMBNs. Because TMBNs are special values of generalized Euler-Zagier-Lerch multiple zeta functions at tuples of non-positive integers, the coefficients can equivalently be described in zeta-value language.

5. CMBNs in the Furusho–Jarossay expansion of $p$0-adic multiple $p$1-function values

For positive integers $p$2, the Furusho–Jarossay expansion gives an expression for

$p$3

as an infinite sum over $p$4, $p$5, and $p$6, in which each summand contains binomial coefficients $p$7, a normalizing factor, a CMBN $p$8, and a CMHV $p$9 subject to combinatorial conditions (Fan, 5 Jul 2026).

The paper’s main theorem then replaces the CMBN coefficient by the explicit TMBN formula and then uses the TMBN–zeta relation. The result is an expansion of the same $L$0-adic family in which the coefficients are written using special values

$L$1

at negative integers of generalized Euler-Zagier-Lerch multiple zeta functions, together with the harmonic-value terms. In the paper’s summary formulation, all coefficients in the expansion of special values of $L$2-adic multiple $L$3-functions in the CMHV basis are now given explicitly as sums of TMBNs, or equivalently, as special values of generalized multiple zeta functions at non-positive integers.

This is the precise role of CMBNs in the $L$4-adic theory: they are the intermediary coefficients in the CMHV expansion, and the paper replaces them by explicit zeta-theoretic data.

6. Special cases and mathematical significance

Two low-depth cases are isolated in the paper. For depth $L$5,

$L$6

For depth $L$7, the special value is written as a sum over $L$8 and $L$9, with coefficients involving

$L$00

together with CMHVs of weight $L$01 (Fan, 5 Jul 2026).

The significance stated in the paper is that this establishes a deep relationship between special values of $L$02-adic multiple $L$03-functions, cyclotomic multiple harmonic values, and values of complex multiple zeta functions at non-positive integers, mediated by explicitly formulated CMBNs. In the paper’s own summary language, CMBNs provide the algebraic bridge between the combinatorics of cyclotomic multiple harmonic sums, values of multiple zeta functions at non-positive integers, and the special values of $L$04-adic multiple $L$05-functions.

A plausible implication is that the explicit formula for CMBNs converts what was formerly a coefficient system defined through cyclotomic harmonic-sum expansions into a coefficient system accessible through Bernoulli-type and zeta-type special values. That reformulation is the key structural contribution of the paper: it makes the coefficient side of the CMHV expansion explicit, finite at the Bernoulli stage, and directly connected to generalized Euler-Zagier-Lerch multiple zeta functions.