Multiple Dirichlet Series Research Articles

Genuine pro-$p$ Iwahori–Hecke algebras, Gelfand–Graev representations, and some applications

We study the Iwahori-component of the Gelfand–Graev representation of a central cover of a split linear reductive group and utilize our results for three applications. In fact, it is advantageous to begin at the pro- p level. Thus to begin we study the structure of a genuine pro- p Iwahori–Hecke algebra, establishing a Bernstein presentation. With this structure theory we first describe the pro- p part of the Gelfand–Graev representation and then the more subtle Iwahori part. For the first application we relate the Gelfand–Graev representation to the metaplectic representation of Sahi–Stokman–Venkateswaran, which conceptually realizes the Chinta–Gunnells action from the theory of Weyl group multiple Dirichlet series. For the second we compute the Whittaker dimension of the constituents of regular unramified principal series representations; for the third we do the same for unitary unramified principal series representations.

Arithmetic progressions of squares and multiple Dirichlet series

Arithmetic progressions of squares and multiple Dirichlet series

Ratios conjecture for quadratic twists of modular L-functions

We develop the L-functions ratios conjecture with one shift in the numerator and denominator in certain ranges for the family of quadratic twist of modular L-functions using multiple Dirichlet series under the generalized Riemann hypothesis.

General multiple Dirichlet series from perverse sheaves

We give an axiomatic characterization of multiple Dirichlet series over the function field Fq(T), generalizing a set of axioms given by Diaconu and Pasol. The key axiom, relating the coefficients at prime powers to sums of the coefficients, formalizes an observation of Chinta. The existence of multiple Dirichlet series satisfying these axioms is proved by exhibiting the coefficients as trace functions of explicit perverse sheaves and using properties of perverse sheaves. The multiple Dirichlet series defined this way include, as special cases, many that have appeared previously in the literature.

The ratios conjecture for real Dirichlet characters and multiple Dirichlet series

Conrey, Farmer and Zirnbauer introduced a recipe to find asymptotic formulas for the sum of ratios of products of shifted L-functions. These ratios conjectures are very powerful and can be used to determine many statistics of L-functions, including moments or statistics about the distribution of zeros. We consider the family of real Dirichlet characters, and use multiple Dirichlet series to prove the ratios conjectures with one shift in the numerator and denominator in some range of the shifts. This range can be improved by extending the family to include non-primitive characters. All of the results are conditional under the Generalized Riemann hypothesis. This extended range is good enough to enable us to compute an asymptotic formula for the sum of shifted logarithmic derivatives near the critical line. As an application, we compute the one-level density for test functions whose Fourier transform is supported in ( − 2 , 2 ) \left (-2,2\right ) , including lower-order terms.

Pseudostarlikeness and Pseudoconvexity of Multiple Dirichlet Series

Let $p\in {\Bbb N}$, $s=(s_1,\ldots,s_p)\in {\Bbb C}^p$, $h=(h_1,\ldots,h_p)\in {\Bbb R}^p_+$, $(n)=(n_1,\ldots,n_p)\in {\Bbb N}^p$ and the sequences $\lambda_{(n)}=(\lambda^{(1)}_{n_1},\ldots,\lambda^{(p)}_{n_p})$ are such that $0<\lambda^{(j)}_1<\lambda^{(j)}_k<\lambda^{(j)}_{k+1}\uparrow+\infty$ as $k\to\infty$ for every $j=1,\ldots,p$. For $a=(a_1,\ldots,a_p)$ and $c=(c_1,\ldots,c_p)$ let $(a,c)=a_1c_1+\ldots+a_pc_p$, and we say that $a>c$ if $a_j> c_j$ for all $1\le j\le p$. For a multiple Dirichlet series \begin{align*}F(s)=e^{(s,h)}+\sum\limits_{\lambda_{(n)}>h}f_{(n)}\exp\{(\lambda_{(n)},s)\}\end{align*} absolutely converges in $\Pi^p_0=\{s:\text{Re}\,s<0\}$, concepts of pseudostarlikeness and pseudoconvexity are introduced and criteria for pseudostarlikeness and the pseudoconvexity are proved. Using the obtained results, we investigated neighborhoods of multiple Dirichlet series, Hadamard compositions, and properties of solutions of some differential equations.

Ratios conjecture for quadratic Hecke L-functions in the Gaussian field

We develop the L-functions ratios conjecture with one shift in the numerator and denominator in certain ranges for the family of quadratic Hecke L-functions in the Gaussian field using multiple Dirichlet series under the generalized Riemann hypothesis. We also obtain an asymptotical formula for the first moment of central values of the same family of L-functions, obtaining an error term of size O(X1/2+ε)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(X^{1/2+\\varepsilon })$$\\end{document}.

Twisted Weyl group multiple Dirichlet series over the rational function field

Weyl group multiple Dirichlet series are Dirichlet series in r complex variables, with analytic continuation to Cr and a group of functional equations isomorphic to the Weyl group of a reduced root system of rank r. Such series may be defined for any global field K, but in the case when K is an algebraic function field they are expected to be, up to a variable change, rational functions in several variables. We verify the rationality of these functions in the case when K=Fq(T), and describe the denominators and support of the numerators.

Wiman’s type inequality for entire multiple Dirichlet series with arbitrary complex exponents

It is proved analogues of the classical Wiman's inequality} for the class $\mathcal{D}$ of absolutely convergents in the whole complex plane $\mathbb{C}^p$ (entire) Dirichlet series of the form $\displaystyle F(z)=\sum\limits_{\|n\|=0}^{+\infty} a_ne^{(z,\lambda_n)}$ with such a sequence of exponents $(\lambda_n)$ that $\{\lambda_n\colon n\in\mathbb{Z}^p\}\subset \mathbb{C}^p$ and $\lambda_n\not=\lambda_m$ for all $n\not= m$. For $F\in\mathcal{D}$ and $z\in\mathbb{C}^p\setminus\{0\}$ we denote 
 $\mathfrak{M}(z,F):=\sum\limits_{\|n\|=0}^{+\infty}|a_n|e^{\Re(z,\lambda_n)},\quad\mu(z,F):=\sup\{|a_n|e^{\mathop{\rm Re}(z,\lambda_n)}\colon n\in\mathbb{Z}^ p_+\},$
 $(m_k)_{k\geq 0}$ is $(\mu_{k})_{k\geq 0}$ the sequence $(-\ln|a_{n}|)_{n\in\mathbb{Z}^p_+}$ arranged by non-decreasing.
 The main result of the paper: Let $F\in \mathcal{D}.$ If $(\exists \alpha > 0)\colon$ $\int\nolimits_{t_0}^{+\infty}t^{-2}{(n_1(t))^{\alpha}}dt<+\infty,$ 
 $n_1(t)\overset{def}=\sum\nolimits_{\mu_n\leq t} 1,\quad t_0>0,$ then there exists a set $E\subset\gamma_{+}(F),$\ such that
 $\tau_{2p}(E\cap\gamma_{+}(F))=\int_{E\cap\gamma_{+}(F)}|z|^{-2p}dxdy\leq C_p, z=x+iy\in\mathbb{C}^p,$ 
 and relation $\mathfrak{M}(z,F)= o(\mu(z,F)\ln^{1/\alpha} \mu(z,F))$ holds as $z\to \infty$\ $(z\in \gamma_R\setminus E)$ for each $R>0$, where
 $\gamma_R=\Big\{z\in\mathbb{C}^p\setminus\{0\}\colon\ K_F(z)\leq R \Big\},\quad K_F(z)=\sup\Big\{\frac1{\Phi_z( t)}\int^{ t}_0 \frac {{\Phi_z}(u)}{u} du\colon\ t \geq t_0\Big\},$ $\gamma(F)=\{z\in\mathbb{C}\colon \ \lim\limits_{t\to +\infty}\Phi_z(t)=+\infty\},\quad \gamma_+(F)=\mathop{\cup}_{R>0}\gamma_R$, $\Phi_z(t)=\frac1{t}\ln\mu(tz,F)$. In general, under the specified conditions, the obtained inequality is exact.

Erdős-Macintyre type theorem’s for multiple Dirichlet series: exceptional sets and open problems

In the paper, we formulate some open problems related to the best description of the values of the exceptional sets in Wiman's inequality for entire functions and in the Erd\H{o}s-Macintyre type theorems for entire multiple Dirichlet series. At the same time, we clarify the statement of one \v{I}.V. Ostrovskii problem on Wiman's inequality. We also prove three propositions and one theorem. On the one hand, in a rather special case, these results give the best possible description of the values of the exceptional set in the Erd\H{o}s-Macintyre-type theorem. On the second hand, they indicate the possible structure of the best possible description in the general case.

Pseudostarlike and pseudoconvex in a direction multiple Dirichlet series

The article introduces the concepts of pseudostarlikeness and pseudoconvexity in the direction of absolutely converges in $\Pi_0=\{s\in\mathbb{C}^p\colon \text{Re}\,s<0\}$, $p\in\mathbb{N},$ the multiple Dirichlet series of the form$$ F(s)=e^{(h,s)}+\sum\nolimits_{\|(n)\|=\|(n^0)\|}^{+\infty}f_{(n)}\exp\{(\lambda_{(n)},s)\}, \quad s=(s_1,...,s_p)\in {\mathbb C}^p,\quad p\geq 1,$$where $ \lambda_{(n^0)}>h$, $\text{Re}\,s<0\Longleftrightarrow (\text{Re}\,s_1<0,...,\text{Re}\,s_p<0)$,$h=(h_1,...,h_p)\in {\mathbb R}^p_+$, $(n)=(n_1,...,n_p)\in {\mathbb N}^p$, $(n^0)=(n^0_1,...,n^0_p)\in {\mathbb N}^p$, $\|(n)\|=n_1+...+n_p$ and the sequences$\lambda_{(n)}=(\lambda^{(1)}_{n_1},...,\lambda^{(p)}_{n_p})$ are such that $0<h_j<\lambda^{(j)}_1<\lambda^{(j)}_k<\lambda^{(j)}_{k+1}\uparrow+\infty$as $k\to+\infty$ for every $j\in\{1,...,p\}$, and $(a,c)=a_1c_1+...+a_pc_p$ for $a=(a_1,...,a_p)$ and $c=(c_1,...,c_p)$. We say that $a>c$ if $a_j\ge c_j$ for all $1\le j\le p$ and there exists at least one $j$ such that $a_j> c_j$. Let ${\bf b}=(b_1,...,b_p)$ and $\partial_{{\bf b}}F( {s})=\sum\limits_{j=1}^p b_j\dfrac{\partial F( {s})}{\partial {s}_j}$ be the derivative of $F$ in the direction ${\bf b}$. In this paper, in particular, the following assertions were obtained: 1) If ${\bf b}>0$ and$\sum\limits_{\|(n)\|=k_0}^{+\infty}(\lambda_{(n)},{\bf b})|f_{(n)}|\le (h,{\bf b})$then $\partial_{{\bf b}}F( {s})\not=0$ in $\Pi_0:=\{s\colon \text{Re}\,s<0\}$, i.e. $F$ is conformal in $\Pi_0$ in the direction ${\bf b}$ (Proposition 1).2) We say that function $F$ is pseudostarlike of the order $\alpha\in [0,\,(h,{\bf b}))$ and the type$\beta >0$ in the direction ${\bf b}$ if$\Big|\frac{\partial_{{\bf b}}F( {s})}{F(s)}-(h, {\bf b})\Big|<\beta\Big|\frac{\partial_{{\bf b}}F( {s})}{F(s)}-(2\alpha-(h, {\bf b}))\Big|,\quad s\in \Pi_0.$Let $0\le \alpha<(h,{\bf b})$ and $\beta>0$. In order that the function $F$ ispseudostarlike of the order $\alpha$ and the type $\beta$ in the direction ${\bf b}> 0$, it is sufficient and in the case, when all $f_{(n)}\le 0$, it is necessary that$\sum\limits_{\|(n)\|=k_0}^{+\infty}\{((1+\beta)\lambda_{(n)}-(1-\beta)h,{\bf b})-2\beta\alpha\}|f_{(n)}|\le 2\beta ((h,{\bf b})-\alpha)$ (Theorem 1).

Some multiple Dirichlet series of completely multiplicative arithmetic functions

Let f_r: \mathbb{N}^r \longrightarrow \mathbb{C} be an arithmetic function of r variables, where r\geq 2. We study multiple Dirichlet series defined by \begin{equation*} D(f_r,s_1,\ldots,s_r)=\sum\limits_{\substack{n_1,\ldots,n_r=1 \\ (n_1,\ldots,n_r)=1}}^{+\infty}\frac{f_r(n_1,\ldots,n_r)}{n_1^{s_1}\cdots n_r^{s_r}}, \end{equation*} where f_r(n_1,\ldots,n_r)=f(n_1)\cdots f(n_r) and f is a completely multiplicative or a specially multiplicative arithmetic function of a single variable. We obtain formulas for these series expressed by infinite products over the primes. We also consider the cases of certain particular completely multiplicative and specially multiplicative functions.

ON THE PARITY RESULT FOR MULTIPLE DIRICHLET SERIES

In this paper, we discuss the parity result for multiple Dirichlet series which contains some special values of multiple zeta functions as special cases, such as the Mordell-Tornheim type of multiple zeta values, zeta values of the root systems and so on. Moreover, we can give an explicit expression in terms of lower series by using the main theorem.

On a multiple Dirichlet series associated to binary cubic forms

In this paper, we construct and study the analytic properties of a multiple Dirichlet series associated to the space of certain binary cubic forms. Using reduction theory, local Euler factor computation, a functional equation and convexity arguments, we prove that this multiple Dirichlet series can be meromorphically continued to the whole of C2.

A Survey on the Theory of Multiple Dirichlet Series with Arithmetical Coefficients as Numerators

A Survey on the Theory of Multiple Dirichlet Series with Arithmetical Coefficients as Numerators

Metaplectic representations of Hecke algebras, Weyl group actions, and associated polynomials

We construct a family of representations of affine Hecke algebras, which depend on a number of auxiliary parameters g_i, and which we refer to as metaplectic representations. We realize these representations as quotients of certain parabolically induced modules, and we apply the method of Baxterization (localization) to obtain actions of corresponding Weyl groups on rational functions on the torus. Our construction both generalizes and provides a conceptual proof of earlier results of Chinta, Gunnells, and Puskas, which had depended on a crucial computer verification. A key motivation is that when the parameters g_i are specialized to certain Gauss sums, the resulting representation and its localization arise naturally in the consideration of p-parts of Weyl group multiple Dirichlet series. In this special case, similar results have been previously obtained in the literature by the study of Iwahori Whittaker functions for principal series of metaplectic covers of reductive p-adic groups. However this technique is not available for generic parameters g_i. It turns out that the metaplectic representations can be extended to the double affine Hecke algebra, where they share many important properties with Cherednik’s basic polynomial representation, which they generalize. This allows us to introduce families of metaplectic polynomials, which depend on the g_i, and which generalize Macdonald polynomials. In this paper we discuss in some detail the situation for type A, which is of considerable interest in algebraic combinatorics. We postpone some of the proofs, as well as a discussion of other types, to the sequel.

Expressions of Schur multiple zeta-functions of anti-hook type by zeta-functions of root systems

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 functions by an analogue of Weyl group multiple Dirichlet series in the sense of Bump et al. By combining with a result of Nakasuji, Phuksuwan and Yamasaki, our theorems yield a new method of finding functional relations among zeta-functions of root systems.

Dirichlet product and the multiple Dirichlet series over function fields

We define the Dirichlet product for multiple arithmetic functions over function fields and consider the ring of the multiple Dirichlet series over function fields. We apply our results to absolutely convergent multiple Dirichlet series and obtain some zero-free regions for them.

English

english

On the crystal graph description of the stable Weyl group multiple Dirichlet series

For a semisimple Lie algebra admitting a good enumeration, we prove a parametrization for the elements in its Weyl group. As an application, we give a coordinate-free comparison between the crystal graph description (when it is known) and the Lie-theoretic description of the Weyl group multiple Dirichlet series in the stable range.