Finite Multiple zeta Values and Finite Euler Sums

TL;DR

This paper investigates the structure and congruences of finite Euler sums, which are finite analogs of Euler sums and multiple zeta values, revealing new relationships and proposing conjectures supported by numerical evidence.

Contribution

It provides systematic structural results on finite Euler sums, relates them to Euler sums, and extends known results from multiple zeta values, introducing new conjectures.

Findings

Structural results on van Hamme type congruences of finite Euler sums

Relations between finite Euler sums and Euler sums

Numerical evidence supporting new conjectures

Abstract

The alternating multiple harmonic sums are partial sums of the infinite series defining the Euler sums which are the alternating version of the multiple zeta value series. In this paper, we present some systematic structural results of the van Hamme type congruences of these sums, collected as finite Euler sums. Moreover, we relate this to the structure of the Euler sums, which generalizes the corresponding result of the multiple zeta values. We also provide a few conjectures with extensive numerical support.