On the evaluations of multiple SS and TT values of the form S(2(−),1,…,1,1(−))S(\overset{{}_{(-)}}{2}, 1, \ldots, 1, \overset{{}_{(-)}}{1}) and T(2(−),1,…,1,1(−))T(\overset{{}_{(-)}}{2}, 1, \ldots, 1, \overset{{}_{(-)}}{1})

Xu, Yan and Zhao showed that in even weight, the multiple $T$ value $T(2, 1, \ldots, 1, \overline{1})$ is a polynomial in $\log(2)$, $\pi$, Riemann zeta values, and Dirichlet beta values. Based on low-weight examples, they conjectured that $\log(2)$ does not appear in the evaluation. We show that their conjecture is correct, and in fact follows largely from various earlier results of theirs. More precisely, we derive explicit formulae for $T(2, 1, \ldots, 1, \overline{1})$ in even weight and $S(2, 1, \ldots, 1, \overline{1})$ in odd weight via generating series calculations. We also resolve another conjecture of theirs on the evaluations of $T(\overline{2}, 1, \ldots, 1, \overline{1})$, $S(\overline{2}, 1, \ldots, 1, 1)$, and $S(\overline{2}, 1, \ldots, 1, \overline{1})$ in even weight, by way of calculations involving Goncharov's theory of iterated integrals and multiple polylogarithms.Comment: 22 page

ζ({{2}^m, 1, {2}^m, 3}^n, {2}^m) / π^(4n+2m(2n+1))) is rational

The cyclic insertion conjecture of Borwein, Bradley, Broadhurst and Lisoněk states that inserting all cyclic shifts of some fixed blocks of 2's into the multiple zeta value ζ(1,3,…,1,3) gives an explicit rational multiple of a power of π . In this paper we use motivic multiple zeta values to establish a non-explicit symmetric insertion result: inserting all possible permutations of some fixed blocks of 2's into ζ(1,3,…,1,3) gives some rational multiple of a power of π

Identities arising from coproducts on multiple zeta values and multiple polylogarithms

In this thesis we explore identities which can be proven on multiple zeta values using the derivation operators $D_r$ from Brown's motivic MZV framework. We then explore identities which occur on multiple polylogarithms by way of the symbol map $\mathcal{S}$, and the multiple polylogarithm coproduct $\Delta$. On multiple zeta values, we consider Borwein, Bradley, Broadhurst, and Lisoněk's cyclic insertion conjecture about inserting blocks of $\{2\}^{a_i}$ between the arguments of $\zeta(\{1,3\}^n)$. We generalise this conjecture to a much broader setting, and give a proof of a symmetrisation of this generalised cyclic insertion conjecture. This proof is by way of the block-decomposition of iterated integrals introduced here, and Brown's motivic MZV framework. This symmetrisation allows us to prove (or to make progress towards) various conjectural identities, including the original cyclic insertion conjecture, and Hoffman's $2\zeta(3,3,\{2\}^n) - \zeta(3,\{2\}^n,1,2)$ identity. Moreover, we can then generate unlimited new conjectural identities, and give motivic proofs of their symmetrisations. We then consider the task of relating weight 5 multiple polylogarithms. Using the symbol map, we determine all of the symmetries and functional equations between depth 2 and between depth 3 iterated integrals with 'coupled-cross ratio' arguments $[\mathrm{cr}(a,b,c,d_1), \ldots, \mathrm{cr}(a,b,c,d_k)]$. We lift the identity for $I_{4,1}(x,y) + I_{4,1}(\frac{1}{x}, \frac{1}{y})$ to an identity holding exactly on the level of the symbol and prove a generalisation of this for $I_{a,b}(x,y)$. Moreover, we further lift the subfamily $I_{n,1}$ to a candidate numerically testable identity using slices of the coproduct. We review Dan's reduction method for reducing the iterated integral $I_{1,1,\ldots,1}$ to a sum in $\leq n-2$ variables. We provide proofs for Dan's claims, and run the method in the case $I_{1,1,1,1}$ to correct Dan's original reduction of $I_{1,1,1,1}$ to $I_{3,1}$ and $I_4$. We can then compare this with another reduction to find $I_{3,1}$ functional equations, and their nature. We then give a reduction of $I_{1,1,1,1,1}$ to $I_{3,1,1}$, $I_{3,2}$ and $I_{5}$, and indicate how one might be able to further reduce to $I_{3,2}$ and $I_5$. Lastly, we use and generalise an idea suggested by Goncharov at weight 4 and weight 5. We find $\mathrm{Li}_n$ terms when certain $\mathrm{Li}_2$, $\mathrm{Li}_3$ and $\mathrm{Li}_4$ functional equations are substituted into the arguments of symmetrisations of $I_{m,1}(x,y)$. By expanding I_{m,1}(\text{\mathrm{Li}_k equation}, \text{\mathrm{Li}_\ell equation}) in two different ways we obtain functional equations for $\mathrm{Li}_5$ and $\mathrm{Li}_6$. We make some suggestions for how this might work at weight 7 and weight 8 giving a potential route to $\mathrm{Li}_7$ and $\mathrm{Li}_8$ functional equations

Evaluation of the multiple zeta values ζ(2,…,2,4,2,…,2)\zeta(2,\ldots,2,4,2,\ldots,2) via double zeta values, with applications to period polynomial relations and to multiple tt values

In studying the depth filtration on multiple zeta values, difficulties quickly arise due to a disparity between it and the coradical filtration. In particular, there are additional relations in the depth graded algebra coming from period polynomials of cusp forms for $SL_2(\mathbb{Z})$. In contrast, a simple combinatorial filtration, the block filtration is known to agree with the coradical filtration, and so there is no similar defect in the associated graded. However, via an explicit evaluation of $\zeta(2,\ldots,2,4,2,\ldots,2)$ as a polynomial in double zeta values, we derive these period polynomial relations as a consequence of an intrinsic symmetry of block graded multiple zeta values in block degree 2. In deriving this evaluation, we find a Galois descent of certain alternating double zeta values to classical double zeta values, which we then apply to give an evaluation of the multiple $t$ values $t(2\ell,2k)$ in terms of classical double zeta values.Comment: 47 pages. Computer readable versions of the full evaluations in Mathematica and pari/gp syntax is included in the arXiv submissio

Clean Single-Valued Polylogarithms

We define a variant of real-analytic polylogarithms that are single-valued and that satisfy ''clean'' functional relations that do not involve any products of lower weight functions. We discuss the basic properties of these functions and, for depths one and two, we present some explicit formulas and results. We also give explicit formulas for the single-valued and clean single-valued version attached to the Nielsen polylogarithms $S_{n,2}(x)$, and we show how the clean single-valued functions give new evaluations of multiple polylogarithms at certain algebraic points.Comment: Special Issue on Algebraic Structures in Perturbative Quantum Field Theory in honor of Dirk Kreimer for his 60th birthda