Multizeta values for function fields: A survey

On multi-poly-Bernoulli–Carlitz numbers

“…Also, Chang-Mishiba and D. S. Thakur concerned v-adic variant ( [10], [19]) and finite variant ( [9], [20]). In this paper, we consider Chang and Mishiba's finite variant ( [9]).…”

“…...,sr),(j1,...,jr )q deg ℘−β d −1 ∏(q deg ℘−β d − 1) = β d >⋯>β0>0 deg ℘−(d+r−l−1)≥β l for each l 1 L deg ℘−β0 ⋯ 1 L deg ℘−β d−1 BC (s1,...,sr ),(j1,...,jr ) q deg ℘−β d −1 ∏(q deg ℘−β d − 1) = deg ℘−(r−1)≥β d >⋯>β0>0 1 L deg ℘−β0 ⋯ 1 L deg ℘−β d−1 BC (s1,...,sr),(j1,...,jr ) q deg ℘−β d −1 ∏(q deg ℘−β d − 1) by putting i l = deg ℘ − β l (d ≥ l ≥ 0), = deg ℘>i0>⋯>i d ≥r−s1,...,sr),(j1,...,jr ) q i d −...,sr),(j1,...,jr ) q i d −Substituting this into the equation(20) and by Γ 1 = 1, we have1 Γ s1 ⋯Γ sr j∈Js a j (θ) deg ℘>i0>⋯>i d ≥r−1 1 L i0 ⋯L i d BC (s1,...,sr ),(j1,...,jr ) q i d −1BC q i d −1 = ζ A k (s) ℘ mod ℘.…”

Zeta-like multiple zeta values in positive characteristic

“…Since their introduction many works have revealed the importance of these values for both their independent interest and for their applications to a wide variety of arithmetic applications, see for example [3,4,5,6,12,22,23,24,25,29,32,33]. We refer the reader to the excellent surveys of Thakur [30,31] for more details and more complete references.…”

On identities for zeta values in Tate algebras

“…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]).…”