Asymptotic relations for weighted finite multiple zeta values

TL;DR

This paper introduces weighted congruences and asymptotic relations for multiple harmonic sums, revealing new algebraic structures and properties of finite multiple zeta values, especially their behavior modulo large powers of primes.

Contribution

It develops a novel framework of weighted congruences and asymptotic relations for multiple harmonic sums, expanding understanding of finite multiple zeta values and their algebraic properties.

Findings

Weighted congruences can hold modulo arbitrarily large powers of p.

Introduction of formal weighted congruences involving infinite terms.

A new algebraic framework classifies weighted congruences and asymptotic relations.

Abstract

Multiple zeta values are real numbers defined by an infinite series generalizing values of the Riemann zeta function at positive integers. Finite truncations of this series are called multiple harmonic sums and are known to have interesting arithmetic properties. When the truncation point is one less than a prime $p$, the mod $p$ values of multiple harmonic sums are called finite multiple zeta values. The present work introduces a new class of congruence for multiple harmonic sums, which we call weighted congruences. These congruences can hold modulo arbitrarily large powers of $p$. Unlike results for finite multiple zeta values, weighted congruences typically involve harmonic sums of multiple weights, which are multiplied by explicit powers of $p$ depending on weight. We also introduce certain formal weighted congruences inolving an infinite number of terms, which we call asymptotic relations. We define a weighted analogue of the finite multiple zeta function, and give an algebraic framework for classifying weighted congruences and asymptotic relations.