[Paper Review] p-adic multiple zeta values and multiple harmonic sums - II : p-adic decompositions of multiple harmonic sums

David Jarossay|arXiv (Cornell University)|Jan 20, 2015

Advanced Mathematical Identities5 references3 citations

TL;DR

This paper studies p-adic multiple zeta values through the lens of multiple harmonic sums with p-adic values, using both geometric and elementary methods to derive decomposition formulas. It introduces a new computational approach for p-adic multiple zeta values that avoids iterated integrals and defines a geometric notion of finite multiple zeta values via Frobenius-invariant paths.

ABSTRACT

We study multiple harmonic sums as functions of their upper bound, with p-adic values ; this study is done in relation with the action of Frobenius on the de Rham fundamental group of P^1 {0,1,\infty}. We obtain results of decomposition, which can be proven both in a geometric way and in a more elementary way. The comparison of the two methods provides a new way to compute explicitly p-adic multiple zeta values, which does not refer directly to iterated integration, and which involves in a natural way the version of p-adic multiple zeta values which expresses the Frobenius-invariant path. The results also lead to the definition of a geometric notion of finite multiple zeta values.

Motivation & Objective

• To analyze multiple harmonic sums as p-adic functions of their upper bound.
• To establish p-adic decomposition formulas for multiple harmonic sums using geometric and elementary methods.
• To provide an alternative computation method for p-adic multiple zeta values that bypasses iterated integration.
• To define a geometric notion of finite multiple zeta values based on Frobenius-invariant paths.
• To unify geometric and elementary approaches to reveal deeper structural properties of p-adic multiple zeta values.

Proposed method

• The study employs the action of Frobenius on the de Rham fundamental group of P^1 minus {0,1,∞} to analyze harmonic sums.
• Decomposition results are derived using both geometric techniques based on the fundamental group and elementary algebraic manipulations.
• The comparison of geometric and elementary methods yields a new algorithm for computing p-adic multiple zeta values.
• The method naturally incorporates the version of p-adic multiple zeta values associated with Frobenius-invariant paths.
• The approach avoids direct reference to iterated integration, offering a more combinatorial and accessible computation route.
• A geometric finite multiple zeta value is defined via the invariance of paths under Frobenius action.

Experimental results

Research questions

• RQ1 How can multiple harmonic sums be decomposed in the p-adic setting using geometric and elementary methods?
• RQ2 What is the relationship between Frobenius-invariant paths and p-adic multiple zeta values?
• RQ3 Can p-adic multiple zeta values be computed without relying on iterated integrals?
• RQ4 How do the geometric and elementary decomposition methods compare in structure and outcome?
• RQ5 What is the natural geometric interpretation of finite multiple zeta values in this context?

Key findings

• A new method for computing p-adic multiple zeta values is developed that does not depend on iterated integration.
• The decomposition of multiple harmonic sums reveals structural patterns consistent with Frobenius action on the fundamental group.
• The comparison of geometric and elementary methods provides a deeper understanding of the underlying algebraic and arithmetic structures.
• A geometric notion of finite multiple zeta values is defined through the invariance of paths under Frobenius.
• The results show that the version of p-adic multiple zeta values tied to Frobenius-invariant paths emerges naturally in the decomposition process.
• The study establishes a bridge between arithmetic geometry and the combinatorics of harmonic sums in the p-adic setting.