We introduce Schur multiple zeta functions which interpolate both the multiple zeta and multiple zeta-star functions of the Euler-Zagier type combinatorially. We first study their basic properties including a region of absolute convergence and the case where all variables are the same. Then, under an assumption on variables, some determinant formulas coming from theory of Schur functions such as the Jacobi-Trudi, Giambelli and dual Cauchy formula are established with the help of Macdonald's ninth variation of Schur functions. Moreover, we investigate the quasi-symmetric functions corresponding to the Schur multiple zeta functions. We obtain the similar results as above for them and, furthermore, describe the images of them by the antipode of the Hopf algebra of quasi-symmetric functions explicitly. Finally, we establish iterated integral representations of the Schur multiple zeta values of ribbon type, which yield a duality for them in some cases.
Inspired by a recent work of M. Nakasuji, O. Phuksuwan and Y. Yamasaki we combine interpolated multiple zeta values and Schur multiple zeta values into one object, which we call interpolated Schur multiple zeta values. Our main result will be a Jacobi-Trudi formula for a certain class of these new objects. This generalizes an analogous result for Schur multiple zeta values and implies algebraic relations between interpolated 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.
We study the Pieri type formulas for the Schur multiple zeta functions along with those for the Schur polynomials. To formulate these formulas, we introduce a new insertion rule for adding boxes in the Young tableaux and obtain the results for the hook type Schur multiple zeta functions. For the proof, we show {certain} extended Jacobi-Trudi formulas for the Schur multiple zeta functions.
We give explicit formulas for the recently introduced Schur multiple zeta values, which generalize multiple zeta(-star) values and which assign to a Young tableaux a real number. In this note we consider Young tableaux of various shapes, filled with alternating entries like a Checkerboard. In particular we obtain new sum representation for odd single zeta values in terms of these Schur multiple zeta values. As a special case we show that some Schur multiple zeta values of Checkerboard style, filled with 1 and 3, are given by determinants of matrices with odd single zeta values as entries.
Nakasuji, Phuksuwan, and Yamasaki defined the Schur multiple zeta values and gave iterated integral expressions of the Schur multiple zeta values of the ribbon type. This paper generalizes their integral expressions to the ones of more general Schur multiple zeta values having constant entries on the diagonals. Furthermore, we also discuss the duality relations for Schur multiple zeta values obtained from the integral expressions.
The algebraic and combinatorial theory of shuffles, introduced by Chen and Ree, is further developed and applied to the study of multiple zeta values. In particular, we establish evaluations for certain sums of cyclically generated multiple zeta values. The boundary case of our result reduces to a former conjecture of Zagier.
We examine "partition zeta functions" analogous to the Riemann zeta function but summed over subsets of integer partitions. We prove an explicit formula for a family of partition zeta functions already shown to have nice properties -- those summed over partitions of fixed length -- which yields complete information about analytic continuation, poles and trivial roots of the zeta functions in the family. Then we present a combinatorial proof of the explicit formula, which shows it to be a zeta function analog of MacMahon's partial fraction decomposition of the generating function for partitions…
In this article, we consider the expressions for content-parametrized Schur multiple zeta-functions in terms of multiple zeta-functions of Euler-Zagier type and their star-variants, or in terms of modified zeta-functions of root systems. First of all, we focus on the Schur multiple zeta-function of hook type. And then, applying the Giambelli formula and induction argument, we obtain the expressions for general content-parametrized Schur multiple zeta-functions.
We investigate relations among Schur multiple zeta functions and zeta-functions of root systems attached to semisimple Lie algebras. Schur multiple zeta functions are defined as sums over semi-standard Young tableaux. Then, assuming the Young tableaux is of anti-hook shape, we show that they can be written in terms of modified zeta-functions of root systems of type $A$. Our proof is quite computational, but we also give a pictorial interpretation of our argument in terms of Young tableaux. It is also possible to understand that one of our theorems gives an expression of Schur multiple zeta fun…
In the study on multiple zeta values, the duality formula is one of the families of basic relations and plays an important role in the investigation of algebraic structure of the space spanned by all multiple zeta values along with the generalized duality formula (so called Ohno relation) obtained by the second author. In this article, we will discuss them for the Schur multiple zeta values which are the values at positive integers of the Schur multiple zeta function introduced by the first author, O. Phukswan and Y. Yamasaki.
Conrey, Farmer, Keating, Rubinstein and Snaith have given a recipe that conjecturally produces, among others, the full moment polynomial for the Riemann zeta function. The leading term of this polynomial is given as a product of a factor explained by arithmetic and a factor explained by combinatorics (or, alternatively, random matrices). We explain how the lower order terms arise, and clarify the dependency of each factor on the exponent $k$ that is considered. We use extensively the theory of symmetric functions and representations of symmetric groups, ideas of Lascoux on manipulations of alp…
The renormalization of MZV was until now carried out by algebraic means. We show that renormalization in general, of the multiple zeta functions in particular, is more than mere convention. We show that simple calculus methods allow us to compute the renormalized values of multiple zeta functions in any dimension for arguments of the form (1,...,1), where the series do not converge. These values happen to be the coefficients of the asymptotic expansion of the inverse Gamma function. We focus on the geometric interpretation of these values, and on the combinatorics their closed form encodes, wh…
The poly-Bernoulli numbers and its relative are defined by the generating series using the polylogarithm series, and we call them type $B$ and $C$, respectively. As a generalization of these poly-Bernoulli numbers, we introduce Schur type poly-Bernoulli numbers and investigate their properties. First, we define a generalization of Arakawa-Kaneko multiple zeta functions and obtain their expression in terms of Schur type Bernoulli numbers. Next, under the restriction to the hook type, we define a generalization of Kaneko-Tsumura multiple zeta functions and obtain similar expression in terms of S…
The sum formulas for multiple zeta(-star) values and symmetric multiple zeta(-star) values bear a striking resemblance. We explain the resemblance in a rather straightforward manner using an identity that involves the Schur multiple zeta values. We also obtain the sum formula for polynomial multiple zeta(-star) values in terms of generating functions, simultaneously generalizing the sum formulas for multiple zeta(-star) values and symmetric multiple zeta(-star) values.
In this work we discuss a parameter $σ$ on weighted $k$-element multisets of $[n]= \{1,\dots ,n\}$. The sums of weighted $k$-multisets are related to $k$-subsets, $k$-multisets, as well as special instances of truncated interpolated multiple zeta values. We study properties of this parameter using symbolic combinatorics. We rederive and extend certain identities for $ζ^{t}_n(\{m\}_k)$. Moreover, we introduce random variables on the $k$-element multisets and derive their distributions, as well as limit laws for $k$ or $n$ tending to infinity.
A large family of relations among multiple zeta values may be described using the combinatorics of shuffle and quasi-shuffle algebras. While the structure of shuffle algebras have been well understood for some time now, quasi-shuffle algebras were only formally studied relatively recently. In particular, Hoffman gives a thorough discussion of the algebraic structure, including a choice of algebra basis, and applies his results to produce families of relations among multiple zeta values and their generalisations. In a recent preprint, Hirose and Sato establish a family of relations coming from …
In 1968 and 1969, Andrews proved two partition theorems of the Rogers-Ramanujan type which generalise Schur's celebrated partition identity (1926). Andrews' two generalisations of Schur's theorem went on to become two of the most influential results in the theory of partitions, finding applications in combinatorics, representation theory and quantum algebra. In a recent paper, the author generalised the first of these theorems to overpartitions, using a new technique which consists in going back and forth between $q$-difference equations on generating functions and recurrence equations on thei…
We find closed-form expressions for the Schur indices of 4d $\mathcal{N}=2^{*}$ super Yang-Mills theory with unitary gauge groups for arbitrary ranks via the Fermi-gas formulation. They can be written as a sum over the Young diagrams associated with spectral zeta functions of an ideal Fermi-gas system. These functions are expressed in terms of the twisted Weierstrass functions, generating functions for quasi-Jacobi forms. The indices lie in the polynomial ring generated by the Kronecker theta function and the Weierstrass functions which contains the polynomial ring of the quasi-Jacobi forms. T…
For a pair of positive integers $n,k$ with $n\geq 2$, in this paper we prove that $$ \sum_{r=1}^k\sum_{|\bfα|=k}{k\choose\bfα} ζ(n\bfα)=ζ(n)^k =\sum^k_{r=1}\sum_{|\bfα|=k} {k\choose\bfα}(-1)^{k-r}ζ^\star(n\bfα), $$ where $\bfα=(α_1,α_2,\ldots,α_r)$ is a $r$-tuple of positive integers. Moreover, we give an application to combinatorics and get the following identity: $$ \sum^{2k}_{r=1}r!{2k\brace r}=\sum^k_{p=1}\sum^k_{q=1}{k\brace p}{k\brace q} p!q!D(p,q), $$ where ${k\brace p}$ is the Stirling numbers of the second kind and $D(p,q)$ is the Delannoy number.
In this short note we will provide a new and shorter proof of the following exotic shuffle relation of multiple zeta values: $$ζ(\{2\}^m \sha\{3,1\}^n)={2n+m\choose m} \frac{π^{4n+2m}}{(2n+1)\cdot (4n+2m+1)!}.$$ This was proved by Zagier when n=0, by Broadhurst when $m=0$, and by Borwein, Bradley, and Broadhurst when m=1. In general this was proved by Bowman and Bradley in \emph{The algebra and combinatorics of shuffles and multiple zeta values}, J. of Combinatorial Theory, Series A, Vol. \textbf{97} (1)(2002), 43--63. Our idea in the general case is to use the method of Borwein et al. to redu…
We estimate asymptotically the fourth moment of the Riemann zeta-function twisted by a Dirichlet polynomial of length $T^{\frac14 - \varepsilon}$. Our work relies crucially on Watt's theorem on averages of Kloosterman fractions. In the context of the twisted fourth moment, Watt's result is an optimal replacement for Selberg's eigenvalue conjecture. Our work extends the previous result of Hughes and Young, where Dirichlet polynomials of length $T^{\frac{1}{11}-\varepsilon}$ were considered. Our result has several applications, among others to the proportion of critical zeros of the Riemann zeta…
There exists an infinite series of ratios by which one can derive the Riemann zeta function $ζ(s)$ from Catalan numbers and central binomial coefficients which appear in the terms of the series. While admittedly the derivation is not deep it does indicate some combinatorial aspect to the Riemann zeta function. But we actually do find also four additional new closed formulas, which include a formula by which one can compute $ζ(s)$ for a countably infinite number of discrete positive values for $s$ where the formula contains Catalan numbers not in infinite series. The Riemann zeta function has a…
In this Ph.D. dissertation (2018, Emory University) we prove theorems at the intersection of the additive and multiplicative branches of number theory, bringing together ideas from partition theory, $q$-series, algebra, modular forms and analytic number theory. We present a natural multiplicative theory of integer partitions (which are usually considered in terms of addition), and explore new classes of partition-theoretic zeta functions and Dirichlet series -- as well as "Eulerian" $q$-hypergeometric series -- enjoying many interesting relations. We find a number of theorems of classical numb…
In this paper, the author constructs a family of algebraic cycles in Bloch's cubical cycle complex over the projective line minus three points which are expected to correspond to multiple polylogarithms in one variable. Elements in this family are in particular equidimensional over the projective line minus three points. In weight greater or equal to $2$, they are naturaly extended as equidimensional cycle over the affine line. This allows to consider their fibers at the point 1 and this is one of the main differences with Gangl, Goncharov and Levin work where generic arguments are imposed for…
Extending the Eulerian functions, we study their relationship with zeta function of several variables. In particular, starting with Weierstrass factorization theorem (and Newton-Girard identity) for the complex Gamma function, we are interested in the ratios of $ζ(2k)/π^{2k}$ and their multiindexed generalization, we will obtain an analogue situation and draw some consequences about a structure of the algebra of polyzetas values, by means of some combinatorics of noncommutative rational series. The same combinatorial frameworks also allow to study the independence of a family of eulerian funct…
The (generalised) Mellin transforms of certain Chebyshev and Gegenbauer functions based upon the Chebyshev and Gegenbauer polynomials, have polynomial factors $p_n(s)$, whose zeros lie all on the `critical line' $\Re\,s=1/2$ or on the real axis (called critical polynomials). The transforms are identified in terms of combinatorial sums related to H. W. Gould's S:4/3, S:4/2 and S:3/1 binomial coefficient forms. Their `critical polynomial' factors are then identified as variants of the S:4/1 form, and more compactly in terms of certain $_3F_2(1)$ hypergeometric functions. Furthermore, we extend t…
