Shuffle regularization for multiple Eisenstein series of level $N$

TL;DR

This paper extends the concept of shuffle regularization from level 1 to arbitrary levels for multiple Eisenstein series, establishing new linear relations among these series.

Contribution

It generalizes shuffle regularization of MES from level 1 to any level N and derives new linear relations among MES of level N.

Findings

Established shuffle regularized MES for arbitrary level N.

Derived linear relations among MES of level N.

Extended known relations from level 1 to higher levels.

Abstract

Bachmann and Tasaka discovered a relationship between multiple Eisenstein series (MES) of level 1 and formal iterated integrals corresponding to multiple zeta value. They also constructed shuffle regularized MES of level 1, which satisfies the shuffle relation that is same as multiple zeta values. In this paper, we expand their results for arbitrary level and give some linear relations among MES of level $N$.