A Weyl group multiple Dirichlet series is a Dirichlet series in several complex variables attached to a root system Phi. The number of variables equals the rank r of the root system, and the series satisfies a group of functional equations isomorphic to the Weyl group W of Phi. In this paper we construct a Weyl group multiple Dirichlet series over the rational function field using n-th order Gauss sums attached to the root system of type A_2. The basic technique is to construct a rational function in r variables invariant under a certain action of W, and use this to build a ``local factor'' of the global series.
Let Phi be a reduced root system of rank r. A Weyl group multiple Dirichlet series for Phi is a Dirichlet series in r complex variables s_1,...,s_r, initially converging for Re(s_i) sufficiently large, which has meromorphic continuation to C^r and satisfies functional equations under the transformations of C^r corresponding to the Weyl group of Phi. Two constructions of such series are available, one based on summing products of n-th order Gauss sums, the second based on averaging a certain group action over the Weyl group. In this paper we study these constructions and the relationship betwee…
Weyl group multiple Dirichlet series are Dirichlet series in $r$ complex variables, with analytic continuation to $\mathbb{C}^r$ 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 $K=\mathbb{F}_q(T)$, and describe the denominators and support of the numerators.
We prove a Weyl-exponent subconvex bound for any Dirichlet $L$-function of cube-free conductor. We also show a bound of the same strength for certain $L$-functions of self-dual $\mathrm{GL}_2$ automorphic forms that arise as twists of forms of smaller conductor.
We prove a Lindelöf-on-average upper bound for the fourth moment of Dirichlet $L$-functions of conductor $q$ along a coset of the subgroup of characters modulo $d$ when $q^*|d$, where $q^*$ is the least positive integer such that $q^2|(q^*)^3$. As a consequence, we finish the previous work of the authors and establish a Weyl-strength subconvex bound for all Dirichlet $L$-functions with no restrictions on the conductor.
We define a multiple Dirichlet series whose group of functional equations is the Weyl group of the affine Kac-Moody root system $\tilde{A}_n$, generalizing the theory of multiple Dirichlet series for finite Weyl groups. The construction is over the rational function field $\mathbb{F}_q(t)$, and is based upon four natural axioms from algebraic geometry. We prove that the four axioms yield a unique series with meromorphic continuation to the largest possible domain and the desired infinite group of symmetries.
We investigate the Shintani zeta functions associated to the prehomogeneous spaces, the example under consideration is the set of $2 \times 2\times 2$ integer cubes. We show that there are three relative invariants under a certain parabolic group action, they all have arithmetic nature and completely determine the equivalence classes. We show that the associated Shintani zeta function coincides with the $A_3$ Weyl group multiple Dirichlet series. Finally, we show that the set of semi-stable integer orbits maps finitely and surjectively to a certain moduli space.
In this paper and its sequel \cite{DPP}, we investigate the precise relationship between the quadratic affine Weyl group multiple Dirichlet series in the sense of \cite{CG1, BD}, and those defined axiomatically by Whitehead \cite{White2} and \cite{White1}. In particular, we show that the axiomatic quadratic Weyl group multiple Dirichlet series of type $D_{\scriptscriptstyle 4}^{\scriptscriptstyle (1)}$ over rational function fields of odd characteristic admits meromorphic continuation to the interior of the corresponding complexified Tits cone. We shall also determine the polar divisor of this…
Let $W$ be the Weyl group of a simply-laced affine Kac-Moody Lie group, excepting $\tilde{A}_n$ for $n$ even. We construct a multiple Dirichlet series $Z(x_1, \ldots x_{n+1})$, meromorphic in a half-space, satisfying a group $W$ of functional equations. This series is analogous to the multiple Dirichlet series for classical Weyl groups constructed by Brubaker-Bump-Friedberg, Chinta-Gunnells, and others. It is completely characterized by four natural axioms concerning its coefficients, axioms which come from the geometry of parameter spaces of hyperelliptic curves. The series constructed this w…
We construct multiple Dirichlet series in several complex variables whose coefficients involve quadratic residue symbols. The series are shown to have an analytic continuation and satisfy a certain group of functional equations. These are the first examples of an infinite collection of unstable Weyl group multiple Dirichlet series in greater than two variables.
We prove a subconvexity bound for the central value L(1/2, chi) of a Dirichlet L-function of a character chi to a prime power modulus q=p^n of the form L(1/2, chi)\ll p^r * q^(theta+epsilon) with a fixed r and theta\approx 0.1645 < 1/6, breaking the long-standing Weyl exponent barrier. In fact, we develop a general new theory of estimation of short exponential sums involving p-adically analytic phases, which can be naturally seen as a p-adic analogue of the method of exponent pairs. This new method is presented in a ready-to-use form and applies to a wide class of well-behaved phases including…
Let Phi be a reduced root system of rank r. A Weyl group multiple Dirichlet series for Phi is a Dirichlet series in r complex variables s_1,...,s_r, initially converging for Re(s_i) sufficiently large, that has meromorphic continuation to C^r and satisfies functional equations under the transformations of C^r corresponding to the Weyl group of Phi. A heuristic definition of such series was given in [2], and they have been investigated in certain special cases in [2-6, 11-14]. In this paper we generalize results in [13] to construct Weyl group multiple Dirichlet series by a uniform method, and …
We formulate a conjecture for the local parts of Weyl group multiple Dirichlet series attached to root systems of type D. Our conjecture is analogous to the description of the local parts of type A series given by Brubaker, Bump, Friedberg, and Hoffstein in terms of Gelfand--Tsetlin patterns. Our conjecture is given in terms of patterns for irreducible representations of even orthogonal Lie algebras developed by Littelmann.
Weyl group multiple Dirichlet series, introduced by Brubaker, Bump, Chinta, Friedberg and Hoffstein, are expected to be Whittaker coefficients of Eisenstein series on metaplectic groups. Chinta and Gunnells constructed these multiple Dirichlet series for all the finite root systems using the method of averaging a Weyl group action on the field of rational functions. In this paper, we generalize Chinta and Gunnells' work and construct Weyl group multiple Dirichlet series for the root systems associated with symmetrizable Kac-Moody algebras, and establish their functional equations and meromorph…
We study the structure of $p$-parts of Weyl group multiple Dirichlet series. In particular, we extend results of Chinta, Friedberg, and Gunnells and show, in the stable case, that the $p$-parts of Chinta and Gunnells agree with those constructed using the crystal graph technique of Brubaker, Bump, and Friedberg. In this vein, we give an explicit recurrence relation on the coefficients of the $p$-parts, which allows us to describe the support of the $p$-parts and address the extent to which they are uniquely determined.
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.
First, we define the multiple Dirichlet product and study the properties of it. From those properties, we obtain a zero-free region of a multiple Dirichlet series and a multiple Dirichlet series expression of the reciprocal of a multiple Dirichlet series.
We prove a Weyl-type subconvexity bound for the central value of the $L$-function of a Hecke-Maass form or a holomorphic Hecke eigenform twisted by a quadratic Dirichlet character, uniform in the archimedean parameter as well as the twisting parameter. A similar hybrid bound holds for quadratic Dirichlet $L$-functions, improving on a result of Heath-Brown. As a consequence of these new bounds, we obtain explicit estimates for the number of Heegner points of large odd discriminant in shrinking sets.
We develop the theory of Weyl group multiple Dirichlet series for root systems of type C. For an arbitrary root system of rank r and a positive integer n, these are Dirichlet series in r complex variables with analytic continuation and functional equations isomorphic to the associated Weyl group. In type C, they conjecturally arise from the Fourier-Whittaker coefficients of minimal parabolic Eisenstein series on an n-fold metaplectic cover of SO(2r+1). For any odd n, we construct an infinite family of Dirichlet series conjecturally satisfying the above analytic properties. The coefficients of …
In this article we study analytic properties of the multiple Dirichlet series associated to additive and Dirichlet characters. For the multiple Dirichlet series associated to additive characters, the meromorphic continuation is established via obtaining translation formulas satisfied by these multiple Dirichlet series. While it seems difficult to obtain such a translation formula for the multiple Dirichlet series associated to Dirichlet characters, we rely on their intrinsic connection with the multiple Dirichlet series associated to additive characters in order to investigate their analytic c…
We consider certain Dirichlet series of Selberg type, constructed from periods of automorphic forms. We study analytic properties of these Dirichlet series and show that they have analytic continuation to the whole complex plane.
B.C. Berndt evaluated special values of the cotangent Dirichlet series. T. Arakawa studied a generalization of the series, or generalized cotangent Dirichlet series, and gave its transformation formulae. In this paper, we establish an elliptic analogue of the generalized cotangent Dirichlet series and give its transformation formulae at some integer arguments. As a corollary, we obtain the transformation formulae of the generalized cotangent Dirichlet series at some integer arguments which are the part of Arakawa's transformation formulae. Those transformation formulae give the special values …
We obtain by simple elementary method, the character average of fourth power of Dirichlet L-series at unity.This result seems to be new.We also obtain character average of second power of Dirichlet L-series at unity,using the power series in second variable of Hurwitz zeta function at unity.
In this paper, we rewrite two forms of an Euler-Ramanujan identity in terms of certain Dirichlet series and derive functional equation of the latter. We also use the Weierstrass-Enneper representation of minimal surfaces to obtain some identities involving these Dirichlet series and one complex parameter.
A multiple Dirichlet series in two variables is constructed as a Mellin transform of a higher order Eisenstein series. It is shown to extend to a meromorphic function and satisfy two independent functional equations.
We present an elementary derivation of the Jacquet-Shalika construction for the exterior square L-function on GL(n), as a classical Dirichlet series in the Fourier coefficients $A(m_1,...,m_{n-1})$.
We study the class numbers of integral binary cubic forms. For each $SL_2(Z)$ invariant lattice $L$, Shintani introduced Dirichlet series whose coefficients are the class numbers of binary cubic forms in $L$. We classify the invariant lattices, and investigate explicit relationships between Dirichlet series associated with those lattices. We also study the analytic properties of the Dirichlet series, and rewrite the functional equation in a self dual form using the explicit relationship.
There exist many explicit evaluations of Dirichlet series. Most of them are constructed via the same approach: by taking products or powers of Dirichlet series with a known Euler product representation. In this paper we derive a result of a new flavour: we give the Dirichlet series representation to solution $f=f(s,w)$ of the functional equation $L(s-wf)=\exp(f)$, where $L(s)$ is the L-function corresponding to a completely multiplicative function. Our result seems to be a Dirichlet series analogue of the well known Lagrange-Bürmann formula for power series. The proof is probabilistic in natur…
As a continuation of the authors and Wakatsuki's previous paper [5], we study relations among Dirichlet series whose coefficients are class numbers of binary cubic forms. We show that for any integral models of the space of binary cubic forms, the associated Dirichlet series satisfies a simple explicit relation to that of the dual other than the usual functional equation. As an application, we write the functional equations of these Dirichlet series in self dual forms.
For a rational valued periodic function, we associate a Dirichlet series and provide a new necessary and sufficient condition for the vanishing of this Dirichlet series specialized at positive integers. This question was initiated by Chowla, and carried out by Okada for a particular infinite sum. Our approach relies on the decomposition of the Dirichlet characters in terms of primitive characters. Using this, we find some new family of natural numbers for which a conjecture of Erdös holds. We also characterize rational valued periodic functions for which the associated Dirichlet series vanishe…
We explore an idea of Conrey and Li of expressing the Selberg trace formula as a Dirichlet series. We describe two applications, including an interpretation of the Selberg eigenvalue conjecture in terms of quadratic twists of certain Dirichlet series, and a formula for an arithmetically weighted sum of the complete symmetric square L-functions associated to cuspidal Maass newforms of squarefree level N>1.
In this paper we study sums of Dirichlet series whose coefficients are terms of the Thue-Morse sequence and variations thereof. We find closed-form expressions for such sums in terms of known constants and functions including the Riemann zeta function and the Dirichlet eta function using elementary methods.
In this article, we obtain the analytic continuation of the multiple shifted Lucas zeta function, multiple Lucas $L$-function associated to Dirichlet characters and additive characters. We then compute a complete list of exact singularities and residues of these functions at these poles. Further, we show the rationality of the multiple Lucas $L$-functions associated with quadratic characters at negative integer arguments.
In this paper, we give criteria for infinitely many sign changes of the coefficients of any Dirichlet series if the coefficients are real numbers. We also provide examples where our criteria are applicable.
Under the generalized Lindelöf Hypothesis in the t- and q-aspects, we bound exponential sums with coefficients of Dirichlet series belonging to a certain class. We use these estimates to establish a conditional result on squares of Hecke eigenvalues at Piatetski-Shapiro primes.
A few years ago, the concept of a D-analogue was introduced as a Dirichlet series analogue for the already known and well researched hypergeometric q-series. The D-analogue of the q-Dixon sum is given here, in the context of seeing a direct comparison with the old Dixon theorem and the q-analogue.
To obtain the Dirichlet series for complex powers of the Riemann zeta function, we define and study the basic properties of a sequence of polynomials that, used as coefficients of the respective terms of the Dirichlet series of the Riemann zeta function in the half plane $x > 1$, produces the required exponential function. Unlike the method described in ([4], p.~278), which requires more advanced knowledge of the relationships between Dirichlet series and multiplicative arithmetic functions, our approach only needs mathematical induction on the total number of prime divisors of $n$, the Dirich…
In this paper, we discuss the parity result for multiple Dirichlet series which contains some special values of multiple zeta functions as special cases, Mordell--Tornheim type of multiple zeta values, zeta values of the root systems and so on. Moreover, we can give explicit expression in terms of lower series by using main theorem.
We present other proofs, generalizations and analogues of the identities concerning multiple Dirichlet series by Tahmi and Derbal (2022). As applications, we obtain asymptotic formulas with remainder terms for certain related sums.
We consider double Dirichlet series associated with arithmetic functions such as the von Mangoldt function, the Möbius function, and so on. We show analytic continuations of them by use of the Mellin-Barnes integral. Furthermore we observe their reverse values at non-positive integer points.
The manuscript reviews Dirichlet Series of important multiplicative arithmetic functions. The aim is to represent these as products and ratios of Riemann zeta-functions, or, if that concise format is not found, to provide the leading factors of the infinite product over zeta-functions. If rooted at the Dirichlet series for powers, for sums-of-divisors and for Euler's totient, the inheritance of multiplicativity through Dirichlet convolution or ordinary multiplication of pairs of arithmetic functions generates most of the results.
We consider a function-field analogue of Dirichlet series associated with the Goldbach counting function, and prove that it can, or cannot, be continued meromorphically to the whole plane. When it cannot, we further prove the existence of the natural boundary of it.
In this rather computational paper, we determine certain representation numbers of ideals in real quadratic number fields explicitly in order to obtain a representation of the associated Dirichlet series in terms of Dirichlet L-functions and a generalized divisor sum. A direct and important consequence is that the Dirichlet series has a meromorphic continuation to the whole complex plane and a simple pole at $s=2$ whose residue can be made explicit in terms of the Dirichlet L-functions and the generalized divisor sum.
Inspired by a famous identity of Ramanujan, we propose a general formula linearizing the convolution of Dirichlet series as the sum of Dirichlet series with modified weights; its specialization produces new identities and recovers several identities derived earlier in the literature, such as the convolution of squares of Bernoulli numbers by A. Dixit and collaborators, or the convolution of Bernoulli numbers by Y. Komori and collaborators.
We attach a certain $n \times n$ matrix $A_n$ to the Dirichlet series $L(s)=\sum_{k=1}^{\infty}a_k k^{-s}$. We study the determinant, characteristic polynomial, eigenvalues, and eigenvectors of these matrices. The determinant of $A_n$ can be understood as a weighted sum of the first $n$ coefficients of the Dirichlet series $L(s)^{-1}$. We give an interpretation of the partial sum of a Dirichlet series as a product of eigenvalues. In a special case, the determinant of $A_n$ is the sum of the Möbius function. We disprove a conjecture of Barrett and Jarvis regarding the eigenvalues of $A_n$.
We investigate a relation between the Mordell-Tornheim type of multiple Dirichlet series and a confluent hypergeometric function. We prove it by applying the Mellin-Barnes integral formula. Also, main results in this paper contain two kinds of functional equations for double $L$-functions as a special case.
The paper considers a method for converting a divergent Dirichlet series into a convergent Dirichlet series by directly converting the coefficients of the original series $1\rightarrowδ_{n}(s)$ for the Riemann Zeta function. In the first part of the paper, we study the properties of the coefficients $δ^*_n$ of a finite Dirichlet series for approximating the Riemann Zeta function on the interval $Δ{H}$. In general, the coefficients $δ^*_n$ of a finite Dirichlet series are complex numbers. The dependence of the coefficients $δ^*_n$ of a finite Dirichlet series on the ordinal number of the coeffi…
