Restricted Sum Formula of Multiple Zeta Values

TL;DR

This paper derives a finite sum formula for restricted multiple zeta values of a specific form, involving binomial coefficients and Bernoulli numbers, applicable for all depths greater than two.

Contribution

It provides a new explicit formula for the sum of multiple zeta values with arguments multiples of four, expanding understanding of their structure.

Findings

Derived a finite sum formula for Q(4n,d) for all d>2

Expressed Q(4n,d) using binomial coefficients, Bernoulli numbers, and Q(4m,2)

Number of terms in the formula is about 3d^2/8, independent of n

Abstract

Let Q(4n,d) be the sum of all multiple zeta values of depth d and weight 4n whose arguments are all multiples of 4. In this paper we derive a formula of Q(4n,d) for all d>2 as a finite sum involving binomial coefficients, Bernoulli numbers and the quantities Q(4m,2). The number of terms is about 3d^2/8 which is independent of n.