In this paper, we study Shen-Shi's colored multizeta values in positive characteristic, which are generalizations of multizeta values in positive characteristic by Thakur. We establish their fundamental properties, that include their non-vanishingness, sum-shuffle relations, t-motivic interpretation and linear independence. In particular, for the linear independence results, we prove that there are no nontrivial k-linear relations among the colored multizeta values of different weights.

In this paper, we study the linear independence of special values, including the positive characteristic analogue of multizeta values, alternating multizeta values and multiple polylogarithms, at algebraic points. Consequently, we establish linearly independent sets of these values with the same weight indices and a lower bound on the dimension of the space generated by depth r > 2 multizeta values of the same weight in positive characteristic.

We introduce multi-poly-Bernoulli-Carlitz numbers, function field analogues of multi-poly-Bernoulli numbers of Imatomi-Kaneko-Takeda. We explicitly describe multi-poly-Bernoulli Carlitz numbers in terms of the Carlitz factorial and the Stirling-Carlitz numbers of the second kind and also show their relationships with function field analogues of finite multiple zeta values.

Abstract. In 1776, L. Euler proposed three methods, called prima methodus, secunda methodus and tertia methodus, to calculate formulae for double zeta values. However, strictly speaking, his last two methods are mathematically incomplete and require more precise reformulation and more sophisticated arguments for their justification. In this paper, we reformulate his formulae, provide their rigorous proofs and also clarify that the formulae can be derived from the extended double shuffle relations. Euler's methodsIn 1776, L. Euler published a celebrated paper Meditations circa singulare serierum genus † in Latin, which translates to 'Meditations about a singular type of series' in English. It is said to be the first publication in history where multiple zeta values (actually only double zeta values) were introduced. In the paper he proposed three methods to calculate certain relations among double zeta values, which he called prima methodus, secunda methodus and tertia methodus. Here we explain the method behind his ideas and point out the steps that would be considered insufficient. Prima methodusIn his paper, he studied the serieswhich he denoted by the unconventional notation (1/z m )(1/y n ) and also the series 1 + 1/2 m + 1/3 m + 1/4 m + · · · which he again denoted by the notation 1/z m . In modern language, they are nothing but the double zeta star valuefor n ∈ Z >0 and m ∈ Z >1 , and the Riemann zeta value ζ (m) := ∞ k=1 1/k m for m ∈ Z >1 , respectively. Here we recall the double zeta value ζ (n, m) :=