Results 1 
7 of
7
Standard Relations of Multiple Polylogarithm Values at Roots of Unity
 DOCUMENTA MATH.
, 2010
"... Let N be a positive integer. In this paper we shall study the special values of multiple polylogarithms at Nth roots of unity, called multiple polylogarithm values (MPVs) of level N. Our primary goal in this paper is to investigate the relations among the MPVs of the same weight and level by using ..."
Abstract

Cited by 10 (8 self)
 Add to MetaCart
(Show Context)
Let N be a positive integer. In this paper we shall study the special values of multiple polylogarithms at Nth roots of unity, called multiple polylogarithm values (MPVs) of level N. Our primary goal in this paper is to investigate the relations among the MPVs of the same weight and level by using the regularized double shuffle relations, regularized distribution relations, lifted versions of such relations from lower weights, and weight one relations which are produced by relations of weight one MPVs. We call relations from the above four families standard. Let d(w, N) be the dimension of the Qvector space generated by all MPVs of weight w and level N. Recently Deligne and Goncharov were able to obtain some lower bound of d(w, N) using the motivic mechanism. We call a level N standard if N = 1, 2, 3 or N = p n for prime p ≥ 5. Our computation suggests the following dichotomy: If N is standard then the standard relations should produce all the linear relations and if further N> 3 then the bound of d(w, N) by Deligne and Goncharov can be improved; otherwise there should be nonstandard relations among MPVs for all sufficiently large weights (depending only on N) and the bound by Deligne and Goncharov may be sharp. We write down some of the nonstandard relations explicitly with good numerical verification. In two instances (N = 4, w = 3, 4) we can rigorously prove these relations by using the octahedral symmetry of {0, ∞, ±1, ± √ −1}. Throughout the paper we provide many conjectures which are strongly supported by computational evidence.
MULTIPOLYLOGS AT TWELFTH ROOTS OF UNITY AND SPECIAL VALUES OF WITTEN MULTIPLE ZETA FUNCTION ATTACHED TO THE EXCEPTIONAL LIE ALGEBRA g2
, 904
"... Abstract. In this note we shall study the Witten multiple zeta function associated to the exceptional Lie algebra g2. Our main result shows that its special values at nonnegative integers can always be expressed as rational linear combinations of the multipolylogs evaluated at 12th roots of unity, ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
(Show Context)
Abstract. In this note we shall study the Witten multiple zeta function associated to the exceptional Lie algebra g2. Our main result shows that its special values at nonnegative integers can always be expressed as rational linear combinations of the multipolylogs evaluated at 12th roots of unity, except for two irregular cases. 1. Introduction and
ALTERNATING EULER SUMS AND SPECIAL VALUES OF WITTEN MULTIPLE ZETA FUNCTION ATTACHED TO so(5)
, 903
"... Abstract. In this note we shall study the Witten multiple zeta function associated to the Lie algebra so(5) defined by Matsumoto. Our main result shows that its special values at nonnegative integers are always expressible by alternating Euler sums. More precisely, every such special value of weight ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
(Show Context)
Abstract. In this note we shall study the Witten multiple zeta function associated to the Lie algebra so(5) defined by Matsumoto. Our main result shows that its special values at nonnegative integers are always expressible by alternating Euler sums. More precisely, every such special value of weight w ≥ 3 is a finite rational linear combination of alternating Euler sums of weight w and depth at most two, except when the only nonzero argument is one of the two last variables in which case ζ(w − 1) is needed. 1.
REDUCIBILITY OF SIGNED CYCLIC SUMS OF MORDELLTORNHEIM ZETA AND LVALUES
, 2009
"... Matsumoto et al. define the MordellTornheim Lfunctions of depth k by ∞ ∑ ∞ ∑ χ1(m1) · · · χk(mk)χk+1(m1 + · · · + mk) LMT(s1,..., sk+1; χ1,...,χk+1): = · · · m s1 1 · · ·msk k (m1 + · · · + mk) sk+1 m1=1 mk=1 for complex variables s1,..., sk+1 and primitive Dirichlet characters χ1,..., ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
Matsumoto et al. define the MordellTornheim Lfunctions of depth k by ∞ ∑ ∞ ∑ χ1(m1) · · · χk(mk)χk+1(m1 + · · · + mk) LMT(s1,..., sk+1; χ1,...,χk+1): = · · · m s1 1 · · ·msk k (m1 + · · · + mk) sk+1 m1=1 mk=1 for complex variables s1,..., sk+1 and primitive Dirichlet characters χ1,...,χk+1. In this paper, we shall show that certain signed cyclic sums of MordellTornheim Lvalues are rational linear combinations of products of multiple Lvalues of lower depths (i.e., reducible). This simultaneously generalizes some results of Subbarao and Sitaramachandrarao, and Matsumoto et al. As a direct corollary, we can prove that for any positive integer n and integer k ≥ 2, the MordellTornheim sums ζMT({n}k, n) is reducible where {n}k denotes the string (n,..., n) with n repeating k times.
INTEGRAL STRUCTURES OF MULTIPLE POLYLOGARITHMS AT ROOTS OF UNITY
"... Abstract. In this paper, for any positive integer N we shall study the special values of multiple polylogarithms at Nth roots of unity, called multiple polylogarithmic values (MPVs) of level N. By standard conjectures linear relations exist only between MPVs of the same weight. LetMPVZ(w,N) be the Z ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. In this paper, for any positive integer N we shall study the special values of multiple polylogarithms at Nth roots of unity, called multiple polylogarithmic values (MPVs) of level N. By standard conjectures linear relations exist only between MPVs of the same weight. LetMPVZ(w,N) be the Zmodule spanned by MPVs of weight w and level N. Our main interest is to investigate for what w and N there exists a basis consisting of MPVs in MPVZ(w,N). In the scope of our investigation this problem always seems to have affirmative answers except for multiple zeta values (level one MPVs) of weight 6 and 7, provided that the conjectural dimensions of MPVQ(w,N) are correct. 1.