$p$-adic multiple $L$-functions and twisted multiple Bernoulli numbers

Abstract: We compute the special values ($p$MLFVs) of the $p$-adic multiple $L$-functions introduced by Furusho, Komori, Matsumoto, and Tsumura at tuples of positive integers. Furusho and Jarossay show that the special values can be expressed as an infinite sum of cyclotomic multiple harmonic values (CMHVs) with coefficients given by cyclotomic multiple Bernoulli numbers (CMBNs). We provide an explicit formula for CMBNs in terms of twisted multiple Bernoulli numbers (TMBNs), which are special values of generalized Euler-Zagier-Lerch type complex multiple zeta functions at tuples of non-positive integers. As a result, we obtain that these $p$MLFVs can be expressed as infinite sums of CMHVs, with coefficients given by the special values of the complex functions at tuples of non-positive integers.

• The paper introduces an explicit formula that expresses p-adic multiple L-function special values as linear combinations of twisted multiple Bernoulli numbers.
• It bridges p-adic analytic objects with complex multiple zeta functions by linking cyclotomic and twisted Bernoulli numbers through combinatorial identities.
• The explicit expressions enable concrete computations in non-archimedean number theory and provide insights into p-adic interpolation and arithmetic properties.

$p$-adic Multiple $L$-Functions and Their Explicit Expression via Twisted Multiple Bernoulli Numbers

Introduction

The paper investigates the structure and explicit evaluation of special values of $p$-adic multiple $L$-functions (abbreviated $p$MLFs), specifically at positive integer arguments. Building on the framework established by Furusho, Komori, Matsumoto, and Tsumura, and further developments by Furusho and Jarossay, the work provides new explicit formulas for the coefficients appearing in the previously known expansions of $p$MLF special values ($p$MLFVs). These results establish direct links between $p$MLFV expansions, cyclotomic multiple Bernoulli numbers (CMBNs), twisted multiple Bernoulli numbers (TMBNs), and special values of Euler-Zagier-Lerch type multiple zeta functions.

$p$-adic Multiple $L$-Functions and Their Expansions

$L$0MLFs, introduced as a natural multivariable $L$1-adic analogue of the classical Kubota-Leopoldt $L$2-adic $L$3-functions, are constructed using $L$4-adic analysis and measures on the ring of $L$5-adic integers. Their special values play a central role in non-archimedean analytic number theory. The evaluation of $L$6MLFVs at positive integer tuples,

$L$7

was previously expressed [CMBN, FKMTp-adic] as convergent infinite sums indexed by combinatorial data. The summands involve cyclotomic multiple harmonic values (CMHVs), and the coefficients are given by cyclotomic multiple Bernoulli numbers (CMBNs).

The main technical obstruction to further explicit understanding of these values has been the complexity of the CMBN coefficients. The present work focuses on resolving this by expressing CMBNs in terms of TMBNs and ultimately as special values of explicit complex multiple zeta functions.

Twisted Multiple Bernoulli Numbers and Their Interpolation Properties

The TMBNs are defined via the expansion coefficients of generating functions involving products over roots of unity: $L$8 They naturally generalize both classical Bernoulli numbers and single-variable twisted Bernoulli numbers. The work of Furusho, Komori, Matsumoto, and Tsumura [FKMTcomplex] connects these coefficients, for roots of unity $L$9, to the special values at non-positive integers of the analytically continued multivariable Euler-Zagier-Lerch zeta function: $p$0 The crucial result is the explicit formula: $p$1 This analytic relation provides the analytic continuation and algebraic control needed for interpreting the coefficients of $p$2MLFV expansions.

Explicit Formula: CMBNs in Terms of TMBNs

The central technical advancement in the paper is an explicit combinatorial formula (Theorem 1, Corollaries for $p$3 and $p$4) expressing any relevant CMBN as a finite sum over TMBNs, with combinatorially determined signs and twisted indices. The formula involves:

• Sums over binary multi-indices parameterizing inclusion-exclusion over the combinatorics of the generating series expansion.
• An explicit identification of indices $p$5 (roots of unity) with products involving the set of twists.
• Binomial coefficients reflecting the possible partitions in the construction.

This explicit formula allows, for any depth $p$6, to rewrite the CMBNs appearing as coefficients in $p$7MLF special value expansions directly as linear combinations of TMBNs. For $p$8, which is the principal case needed in the expansion of $p$9MLFV’s, the formula substantially simplifies.

Reformulation of the $L$0MLFV Expansion

By substituting the explicit expressions of CMBNs in terms of TMBNs into the known infinite series expansions for $L$1MLFVs and applying the relation between TMBNs and zeta values, the author obtains a full, explicit representation of $L$2MLF special values:

• Each coefficient in the expansion is expressed as a special value at non-positive integers of an Euler-Zagier-Lerch type multiple zeta function.
• The sum remains indexed by multi-indices, combinatorial triplets $L$3 (encoding various shuffling and partitioning of summation indices), and roots of unity for the twists.
• The summands contain CMHVs with combinatorially determined depth and weight.

Concrete Low-Degree Cases

The structure is completely transparent in small cases:

• Depth 1 ($L$4): The expansion becomes, for positive integer $L$5,

$L$6

so the coefficients are special values of Hurwitz-type zeta functions at non-positive integers, and the summands are twisted cyclotomic harmonic values.

• Depth 2: The author provides the full expansion, which incorporates both depth-2 and symmetrized combinations, and further demonstrates the combinatorial symmetry and cancellation structure inherent in the expansion.

Implications

From a theoretical perspective, these explicit formulas:

• Completely resolve the expansion of $L$7MLF special values in terms of data from complex analytic multiple zeta functions, thereby connecting $L$8-adic analytic objects to values of (desingularized) multivariable zeta functions at special points.
• Provide new combinatorial and algebraic relations among TMBNs, CMBNs, and the values of CMHVs.
• Give a computationally accessible description of $L$9MLFVs, reducing questions about their algebraicity, integrality, distribution, and possible relations to questions about the special values of well-understood analytic objects.
• Strengthen the bridge between $p$0-adic multiple zeta values and their complex counterparts, informing conjectures about their motivic and Galois-theoretic properties.

On the practical side, these results enable:

• Explicit computations of special values of $p$1-adic multiple $p$2-functions, which are central in Iwasawa theory, arithmetic geometry, and the study of generalized Euler systems.
• Further study of the $p$3-adic interpolation properties and congruences for these special values, since all combinatorial and analytic data are now made explicit.

Perspectives for Future Research

The described explicit framework opens several research avenues:

• Investigation of $p$4-adic and global functional equations for $p$5MLFs, exploiting the explicit connection with TMBNs and zeta values.
• Extension to more general coefficients, including weight spaces, or to $p$6-adic analogues of zeta and polylogarithm functions with additional algebraic or motivic structures.
• Study of congruences and algebraic relations among the special values for varying $p$7, possibly leading to new generalizations of Kummer-type congruences or $p$8-adic Beilinson conjectures.
• Application to the explicit construction and understanding of Euler systems in the $p$9-adic and higher-dimensional Iwasawa theory framework.

Conclusion

The paper provides a complete and explicit formulation of the special values of $p$0-adic multiple $p$1-functions at positive integers in terms of twisted multiple Bernoulli numbers and, through them, special values of generalized Euler-Zagier-Lerch multiple zeta functions. This bridges the gap between non-archimedean and complex analytic number theory, giving accessible formulas for further study of arithmetic, analytic, and motivic properties of $p$2MLFs and related arithmetic invariants.

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References:

(2607.04086)