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It is known that multiple zeta values can be written in terms of certain iterated log-sine integrals. Conversely, we evaluate iterated log-sine integrals in terms of multiple zeta values and multiple polylogarithms in this paper. We also suggest some conjectures on multiple zeta values, multiple Clausen values, multiple Glaisher values and iterated log-sine integrals. [ABSTRACT FROM AUTHOR]

Classical multiple zeta values can be viewed as iterated integrals of the differentials d t t , d t 1 − t from 0 to 1. In this paper, we reprove Brown's theorem: For a i , b i , c i j ∈ ℤ , the iterated integral of the form ∫ ⋯ ∫ 0 < t 1 < ⋯ < t N < 1 ∏ i t i a i (1 − t i) b i ∏ i < j (t j − t i) c i j d t 1 ⋯ d t N is a ℚ -linear combination of multiple zeta values of weight ≤ N if convergent. What is more, we show that if p i (t) , 1 ≤ i ≤ N , are in a ℚ [ t , 1 / t , 1 / (1 − t) ] -algebra generated by multiple polylogarithms and their dual, and if q i j (t) , 1 ≤ i < j ≤ N , are in a ℚ [ t , 1 / t ] -algebra generated by logarithm, then the iterated integral ∫ ⋯ ∫ 0 < t 1 < ⋯ < t N < 1 ∏ i p i (t i) ∏ i < j q i j (t j − t i) d t 1 ⋯ d t N is a ℚ -linear combination of multiple zeta values. As an application of our main results, we show that the coefficients of the Taylor expansions of the Selberg integrals ∫ ⋯ ∫ 0 < t 1 < ⋯ < t N < 1 f ∏ i t i α i (1 − t i) β i ∏ i < j (t j − t i) γ i j d t 1 ⋯ d t N (with respect to α i , β i , γ i j ) at the integral points in some product of right half complex plane are ℚ -linear combinations of multiple zeta values for any f ∈ ℚ [ t i , t i − 1 , (t i − t j) − 1 | 1 ≤ i ≤ N , 1 ≤ i < j ≤ N ]. This statement generalizes Terasoma's original result. [ABSTRACT FROM AUTHOR]

In this paper, we show that the iterated integrals on products of one variable multiple polylogarithms from 0 to 1 are actually in the algebra of multiple zeta values if they are convergent. In the divergent case, we define the regularized iterated integrals from 0 to 1. By the same method, we show that the regularized iterated integrals are also in the algebra of multiple zeta values. As an application, we give new series representations for multiple zeta values and calculate some interesting examples of iterated integrals. [ABSTRACT FROM AUTHOR]

Polylogarithms are those multiple polylogarithms that factor through a certain quotient of the de Rham fundamental group of the thrice punctured line known as the polylogarithmic quotient. Building on work of Dan-Cohen, Wewers, and Brown, we push the computational boundary of our explicit motivic version of Kim's method in the case of the thrice punctured line over an open subscheme of Spec  ℤ. To do so, we develop a greatly refined version of the algorithm of Dan-Cohen tailored specifically to this case, and we focus attention on the polylogarithmic quotient. This allows us to restrict our calculus with motivic iterated integrals to the so-called depth-one part of the mixed Tate Galois group studied extensively by Goncharov. We also discover an interesting consequence of the symmetry-breaking nature of the polylog quotient that forces us to symmetrize our polylogarithmic version of Kim's conjecture. In this first part of a two-part series, we focus on a specific example, which allows us to verify an interesting new case of Kim's conjecture. [ABSTRACT FROM AUTHOR]

We give a new proof of the duality of multiple zeta values, which makes no use of the iterated integrals. The same method is also applicable to Ohno's relation for (q -)multiple zeta values. [ABSTRACT FROM AUTHOR]

In this paper, we prove a family of identities among multiple zeta values, which contains as a special case a conjectural identity of Hoffman. We use the iterated integrals on ℙ 1 ∖ { 0 , 1 , ∞ , z } for our proof. [ABSTRACT FROM AUTHOR]

The evaluation of iterated primitives of powers of logarithms is expressed in closed form. The expressions contain polynomials with coefficients given in terms of the harmonic numbers and their generalizations. The logconcavity of these polynomials is established. [ABSTRACT FROM AUTHOR]