Bachmann–Kühn’s brackets and multiple zeta values at level N

Abstract

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 Kühn 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 D = q d dq on MDFs vanishes on the MZV side, which gives rise to many nontrivial Q-linear relations. In a sequel to this paper, we plan to investigate the nature of these relations.

Restricted sum formula of alternating Euler sums

“…In [14] Kaneko and Tasaka considered the double zeta values and double Eisenstein series at level 2. Yuan and and the author [22] generalized these to level N by considering the subseries of a MZV series where the summation indices run through any fixed congruence class modulo N . Similarly, the MMVs are clearly the multiple variable version at level 2 and therefore can be extended to level N .…”

“…Similarly, the MMVs are clearly the multiple variable version at level 2 and therefore can be extended to level N . In [22] we defined their two ways of regularization and the corresponding double shuffle relations so that it is conceivable that many results in this paper may be extended to arbitrary levels using the results on colored MZVs.…”

Explicit Relations Between Kaneko-Yamamoto Type Multiple Zeta Values and Related Variants

“…Remark 4.7. It is possible to give an induction proof of Theorem 4.4 using the regularized values of MMVs as defined by [15,Definition 3.2]. However, the general formula for the integral of L(k, 1; x) would be implicit.…”