多重ゼータ値の合流関係式 (Algebraic Number Theory and Related Topics 2017)

Abstract

This article is a research announcement of my upcoming joint paper with Minoru Hirose on a certain class of Q-linear relations among the multiple zeta values (MZVs), which we call confluence relations. As is well known, MZVs are iterated integrals of meromorphic one-forms dt/t and dt/t-1 on a projective line. Here we consider more general iterated integrals of three different one-forms dt/t, dt/t-1 and dt/t-z and define the confluence relations as limits as z → 1 of Q-linear relations among these iterated integrals. At first, we define standard relations among the iterated integrals which naturally arise by regarding them as functions of z and thus using their differential structure with respect to z, and then we consider their limits as z → 1. The confluence relations seem to give a very rich family of Q-linear relations among MZVs and we even propose a conjecture that they exhaust all the Q-linear relations among MZVs. As a good reason for our conjecture, we prove that the confluence relations imply the extended double shuffle relations (also the duality realtion). A small table up to weight 4 of the confluence relations is given at the end.