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One of the most interesting formulas for multiple zeta values is the sum formula proved by Granville and Zagier independently in 1990s. Many variations and generalizations of it have been found since then. In this paper, we will provide a uniform approach to proving these results using the regularized double shuffle relations of Euler sums. We summarize these formulas including many new ones with three tables in the Appendix at the end of the paper. [ABSTRACT FROM AUTHOR]

We give explicit expressions for MacMahon's generalized sums-of-divisors q-series A r and C r by relating them to (odd) multiple Eisenstein series. Recently, these sums-of-divisors have been studied in the context of quasimodular forms, vertex algebras, N = 4 SU(N) Super–Yang–Mills theory, and the study of congruences of partitions. We relate them to a broader mathematical framework and give explicit expressions for both q-series in terms of Eisenstein series and their odd variants. [ABSTRACT FROM AUTHOR]

Multiple zeta values at non-positive integers depend on its limiting process. In this paper, explicit and recurrence formulas to evaluate multiple zeta values at non-positive integers for various limiting processes are given. As applications of them, a classification of multiple zeta values for ordered limits and relationship between multiple zeta values for directional limits and ordered limits are showed. Further, an alternative proof of the conjecture raised by Akiyama, Egami and Tanigawa is also given. [ABSTRACT FROM AUTHOR]

In this paper we study the absolute convergence of general multiple Dirichlet series defined by ∑ m 1 = 1 ∞ ∑ m 2 = 1 ∞ ⋯ ∑ m r = 1 ∞ a 1 (m 1) a 2 (m 2) ⋯ a r (m r) m 1 s 1 (m 1 + m 2) s 2 ⋯ (m 1 + m 2 + ⋯ + m r) s r , where a j (1 ≤ j ≤ r) are arithmetic functions. In particular we completely determine the region of absolute convergence under certain conditions on the arithmetic functions. [ABSTRACT FROM AUTHOR]

We study a refinement of the symmetric multiple zeta value, called the t-adic symmetric multiple zeta value, by considering its finite truncation. More precisely, two kinds of regularizations (harmonic and shuffle) give two kinds of the t-adic symmetric multiple zeta values, thus we introduce two kinds of truncations correspondingly. Then we show that our truncations tend to the corresponding t-adic symmetric multiple zeta values, and satisfy the harmonic and shuffle relations, respectively. This gives a new proof of the double shuffle relations for t-adic symmetric multiple zeta values, first proved by Jarossay. In order to prove the shuffle relation, we develop the theory of truncated t-adic symmetric multiple zeta values associated with 2-colored rooted trees. Finally, we discuss a refinement of Kaneko–Zagier's conjecture and the t-adic symmetric multiple zeta values of Mordell–Tornheim type. [ABSTRACT FROM AUTHOR]