Some new proof of the sum formula and restricted sum formula

Abstract

The sum formula is a basic identify of multiple zeta values that expresses a Riemann-zeta value as a homogeneous sum of multiple zeta values of given depth and weight. This formula was already known to Euler in the depth two case. Conjectured in the early 1990s, for higher depth and then proved by Granville and Zagier independently. Restricted sum formula was given in Eie \cite{2}. In this paper, we present some new proofs of those formulas.