In this paper we shall define the analytic continuation of the multiple (Euler-Riemann-Zagier) zeta functions of depth $d$ : \[ $\zeta (s_1,\dots ,s_d):= \sum _{0>n_1 > n_2>\cdots >n_d} \frac {1}{n_1^{s_1}n_2^{s_2}\cdots n_d^{s_d}},$ \] where $\operatorname {Re}(s_d)>1$ and $\sum _{j=1}^d\operatorname {Re}(s_j)>d$ . We shall also study their behavior near the poles and pose some open problems concerning their zeros and functional equations at the end.

Let N be a positive integer. In this paper we shall study the special values of multiple polylogarithms at N th 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 \mathbb Q -vector 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\ge 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 non-standard 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 non-standard 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,\infty,\pm1,\pm\sqrt{-1}} . Throughout the paper we provide many conjectures which are strongly supported by computational evidence.

Let s1,⋯,sd be d positive integers and define the multiple t-values of depth d by t(s1,⋯,sd)=∑n1>⋯>nd≥11(2n1-1)s1⋯(2nd-1)sd, which is equal to the multiple Hurwitz-zeta value 2-wζ(s1,⋯,sd;-12,⋯,-12) where w=s1+⋯+sd is called the weight. For d≤n , let T(2n,d) be the sum of all multiple t-values with even arguments whose weight is 2n and whose depth is d. In 2011, Shen and Cai gave formulas for T(2n,d) for d≤5 in terms of t(2n) , t(2)t(2n-2) and t(4)t(2n-4) . In this short note we generalize their results to arbitrary depth by using the theory of symmetric functions established by Hoffman (2012).