Multiple Divisor Functions and Multiple Zeta Values at Levle N

Multiple zeta values (MZVs) are generalizations of Riemann zeta values at positive integers to multiple variable setting. These values can be further generalized to level $N$ multiple polylog values by evaluating multiple polylogs at $N$-th roots of unity. In this paper, we consider another level $N$ generalization by restricting the indices in the iterated sums defining MZVs to congruences classes modulo $N$, which we call the MZVs at level $N$. The goals of this paper are two-fold. First, we shall lay down the theoretical foundations of these values such as their regularizations and double shuffle relations. Second, we will generalize the multiple divisor functions (MDFs) defined by Bachman and K\"uhn to arbitrary level $N$ and study their relations to MZVs at level $N$. These functions are all $q$-series and similar to MZVs, they have both weight and depth filtrations. But unlike that of MZVs, the product of MDFs usually has mixed weights; however, after projecting to the highest weight we can obtain an algebra homomorphism from MDFs to MZVs. Moreover, the image of the derivation $\mathfrak{D}=q\frac{d}{dq}$ on MDFs vanishes on the MZV side, which gives rise to many nontrivial $\mathbb{Q}$-linear relations. In a sequel to this paper, we plan to investigate the nature of these relations.