On twisted A-harmonic sums and Carlitz finite zeta values

Abstract

In this paper, we study various twisted A-harmonic sums, named following the seminal log-algebraicity papers of G. Anderson. These objects are partial sums of new types of special zeta values introduced by the first author and linked to certain rank one Drinfeld modules over Tate algebras in positive characteristic by Anglès, Tavares Ribeiro and the first author. We prove, by using techniques introduced by the second author, that various infinite families of such sums may be interpolated by polynomials, and we deduce, among several other results, properties of analogues of finite zeta values but inside the framework of the Carlitz module. In the theory of finite multi-zeta values in characteristic zero, finite zeta values are all zero. In the Carlitzian setting, there exist non-vanishing finite zeta values, and we study some of their properties in the present paper.

On identities for zeta values in Tate algebras

“…For q large enough we then compute explicitly this expression of B s (see Theorem 3.1), which implies immediately the desired estimation of its weight (see Section 4). We mention that the proof of Theorem 3.1 is of combinatorial nature and that combinatorial properties of B s have already had important applications in function field arithmetic (see [4,20,27] for more details).…”

“…A few explicit examples of the polynomials B s are given in [4,7] (see also [27]). We need to introduce some more notation.…”

“…Since their introduction various works have revealed the importance of these zeta values for both their proper interest and their applications to values of the Goss L-functions, characteristic p multiple zeta values, Anderson's log-algebraicity identities, Taelman's units, and Drinfeld modular forms in Tate algebras (see for example [4,6,7,8,9,11,12,14,15,27,28,32]). We should mention that generalizations of these zeta values to various settings have been also conducted (see for example [2,3,22,23,24]).…”

A sum-shuffle formula for zeta values in Tate algebras

“…This formula was first observed by Rudolph Perkins. We can deduce it from the formula (5) of [9] (see also the preprint [10]), where it is proved, for Σ = {1, . .…”

Twisted characteristic p zeta functions

“…Such an analytic class number formula has been generalized in [16] to a larger class of L-series (in particular for L(n; t; z), for n ≥ 1). We also refer the reader to [5], [6], [23], [30], [31], [32], [33], [34], [37] for various arithmetic and analytic properties of the series L(n; t; z), n ∈ Z. Now, let P be a monic irreducible polynomial in A of degree d. Let C P be the completion of an algebraic closure K P of the P -adic completion of K. Let A P be the valuation ring of K P .…”