arXiv subjects

Steven Charlton

Publications and source records attributed to Steven Charlton.

31 records · Page 2Linked to original sources

arXiv2022

On two conjectures of Sun concerning Apéry-like series

In this paper, we shall prove two conjectures of Z.-W. Sun concerning Apéry-like series. One of the series is alternating whereas the other one is not. Our main strategy is to convert the series (resp.~the alternating series) to log-sine-cosine (resp.~log-sinh-cosh) integrals. Then we express all these integrals in terms of single-valued Bloch-Wigner-Ramakrishnan-Wojtkowiak-Zagier polylogarithms. The conjectures then follow from a few highly non-trivial functional equations of the polylogarithms of weight $3$ and $4$.

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arXiv2022

Explicit formulas for Grassmannian polylogarithms

We give new explicit formulas for Grassmannian and Aomoto polylogarithms in terms of iterated integrals, for arbitrary weight. We also explicitly reduce the Grassmannian polylogarithm in weight 4 and in weight 5 each to depth 2. Furthermore, using this reduction in weight 4 we obtain an explicit, albeit complicated, form of the so-called 4-ratio, which gives an expression for the Borel class in continuous cohomology of $\mathrm{GL}_4$ in terms of $\mathrm{Li}_4$.

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arXiv2022

Symmetry results for multiple $t$-values

For a composition $I$ whose first part exceeds 1, we can define the multiple $t$-value $t(I)$ as the sum of all the terms in the series for the multiple zeta value $ζ(I)$ whose denominators are odd. In this paper we show that if $I$ is composition of $n\ge 3$, then $t(I)=(-1)^{n-1}t(\bar I)$ mod products, where $\bar I$ is the reverse of $I$, and both sides are suitably regularized when $I$ ends in 1. This result is not true for multiple zeta values, though there is an argument-reversal result that does hold for them (and for multiple $t$-values as well). We actually prove a more general version of this result, and then use it to establish explicit formulas for several classes of multiple $t$-values and interpolated multiple $t$-values.

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arXiv2021

On the evaluation of the alternating multiple $t$ value $t(\{\overline{1}\}^a, 1, \{\overline{1}\}^b)$

We prove an evaluation for the stuffle-regularised multiple $t$ value $ t^{\ast,V}(\{\overline{1}\}^a, 1, \{\overline{1}\}^b) $ in terms of $ \log(2) $, $ ζ(k) $ and $ β(k) $. This arises by evaluating the corresponding generating series using the Evans-Stanton/Ramanujan asymptotics of a zero-balanced hypergeometric function $ {}_3F_2 $, and an evaluation established by Li in an alternative approach to Zagier's evaluation of $ ζ(\{2\}^a, 3, \{2\}^b) $. We end with some discussion and conjectures on possible motivic applications.

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arXiv2021

On motivic multiple $t$ values, Saha's basis conjecture, and generators of alternating MZV's

We give an evaluation for the stuffle-regularised $t^{\ast,V}(\{2\}^a,1,\{2\}^b)$ as a polynomial in single-zeta values, $\log(2)$ and $V$. We then apply this to establish some linear independence results of certain sets of motivic multiple $t$ values. In particular, we prove the elements of Saha's conjectural basis are linearly independent, on the motivic level, and that the (suitably regularised) elements $t^\mathfrak{m}(\{1,2\}^\times)$ form a basis for both the (extended) motivic MtV's and the alternating MZV's.

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arXiv2021

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.

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arXiv2020

Functional equations of polygonal type for multiple polylogarithms in weights 5, 6 and 7

We present new functional equations in weights 5, 6 and 7 and use them for explicit depth reduction of multiple polylogarithms. These identities generalize the crucial identity $\mathbf{Q}_4$ from the recent work of Goncharov and Rudenko that was used in their proof of the weight 4 case of Zagier's Polylogarithm Conjecture.

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arXiv2019

Generalized Jacobi-Trudi determinants and evaluations of Schur multiple zeta values

We present new determinant expressions for regularized Schur multiple zeta values. These generalize the known Jacobi-Trudi formulae and can be used to quickly evaluate certain types of Schur multiple zeta values. Using these formulae we prove that every Schur multiple zeta value with alternating entries in 1 and 3 can be written as a polynomial in Riemann zeta values. Furthermore, we give conditions on the shape, which determine when such Schur multiple zetas are polynomials purely in odd or in even Riemann zeta values.

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arXiv2019

On functional equations for Nielsen polylogarithms

We derive new functional equations for Nielsen polylogarithms. We show that, when viewed modulo $\mathrm{Li}_5$ and products of lower weight functions, the weight $5$ Nielsen polylogarithm $S_{3,2}$ satisfies the dilogarithm five-term relation. We also give some functional equations and evaluations for Nielsen polylogarithms in weights up to 8, and general families of identities in higher weight.

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arXiv2018

Analogues of cyclic insertion type identities for multiple zeta star values

We prove an identity for multiple zeta star values, which generalises some identities due to Imatomi, Tanaka, Tasaka and Wakabayashi. This identity gives an analogue of cyclic insertion type identities, for multiple zeta star values, and connects the block decomposition with Zhao's generalised 2-1 formula.

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arXiv2017

The alternating block decomposition of iterated integrals, and cyclic insertion on multiple zeta values

The cyclic insertion conjecture of Borwein, Bradley, Broadhurst and Lisoněk states that by inserting all cyclic permutations of some initial blocks of 2's into the multiple zeta value $ ζ(1,3,\ldots,1,3) $ and summing, one obtains an explicit rational multiple of a power of $ π$. Hoffman gives a conjectural identity of a similar flavour concerning $ 2 ζ(3,3,\{2\}^m) - ζ(3,\{2\}^m,(1,2)) $. In this paper we introduce the 'generalised cyclic insertion conjecture', which we describe using a new combinatorial structure on iterated integrals -- the so-called alternating block decomposition. We see that both the original BBBL cyclic insertion conjecture, and Hoffman's conjectural identity, are special cases of this 'generalised' cyclic insertion conjecture. By using Brown's motivic MZV framework, we establish that some symmetrised version of the generalised cyclic insertion conjecture always holds, up to a rational; this provides some evidence for the generalised conjecture.

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arXiv2017

A review of Dan's reduction method for multiple polylogarithms

In this paper we will give an account of Dan's reduction method for reducing the weight $ n $ multiple logarithm $ I_{1,1,\ldots,1}(x_1, x_2, \ldots, x_n) $ to an explicit sum of lower depth multiple polylogarithms in $ \leq n - 2 $ variables. We provide a detailed explanation of the method Dan outlines, and we fill in the missing proofs for Dan's claims. This establishes the validity of the method itself, and allows us to produce a corrected version of Dan's reduction of $ I_{1,1,1,1} $ to $ I_{3,1} $'s and $ I_4 $'s. We then use the symbol of multiple polylogarithms to answer Dan's question about how this reduction compares with his earlier reduction of $ I_{1,1,1,1} $, and his question about the nature of the resulting functional equation of $ I_{3,1} $. Finally, we apply the method to $ I_{1,1,1,1,1} $ at weight 5 to first produce a reduction to depth $ \leq 3 $ integrals. Using some functional equations from our thesis, we further reduce this to $ I_{3,1,1} $, $ I_{3,2} $ and $ I_5 $, modulo products. We also see how to reduce $ I_{3,1,1} $ to $ I_{3,2} $, modulo $ δ$ (modulo products and depth 1 terms), and indicate how this allows us to reduce $ I_{1,1,1,1,1} $ to $ I_{3,2} $'s only, modulo $ δ$.

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arXiv2013

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.

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