Multiple Dirichlet series and automorphic forms

Given a root system Φ of rank r and a global field F containing the n-th roots of unity, it is possible to define a Weyl group multiple Dirichlet series whose coefficients are n-th order Gauss sums. It is a function of r complex variables, and it has meromorphic continuation to all of C, with functional equations forming a group isomorphic to the Weyl group of Φ. Weyl group multiple Dirichlet series and their residues unify many examples that have been studied previously in a case-bycase basis, often with applications to analytic number theory. (Examples may be found in the final section of the paper.) We believe these Weyl group multiple Dirichlet series are fundamental objects. The goal of this paper is to define these series for any such Φ and F , and to indicate how to study them. We will note the following points. • The coefficients of the Weyl group multiple Dirichlet series are multiplicative, but the multiplicativity is twisted , so the Dirichlet series is not an Euler product. • Due to the multiplicativity, description of the coefficients reduces to the case where the parameters are powers of a single prime p. There are only finitely many such coefficients (for given p). • In the “stable case” where n is sufficiently large (depending on Φ), the number of nonzero coefficients in the p-part is equal to the order of the Weyl group. Indeed, these nonzero coefficients are parametrized in a natural way by the Weyl group elements. • The p-part coefficient parametrized by a Weyl group element w is a product of l(w) Gauss sums, where l is the length function on the Weyl group. We note a curious similarity between this description and the coefficients of the generalized theta series on the n-fold cover of GL(n) and GL(n − 1); these coefficients are determined in Kazhdan and Patterson [17] and discussed further in Patterson [21]. See Bump and Hoffstein [11] or Hoffstein [15] for a “classical” description of these coefficients. The noted similarity means that the complete Mellin transform of the theta function would be a multiple Dirichlet series resembling our An+1 multiple Dirichlet series. There is no a priori reason that we are

General multiple Dirichlet series from perverse sheaves

Friedberg, Hoffstein and Lieman have constructed two related multiple Dirichlet series from quadratic and higher-order L-functions and Gauss sums. We compute these multiple Dirichlet series explicitly in the case of the rational function field. This is done by utilizing the functional equation of the L-functions and the functional equation relating the two multiple Dirichlet series. We also point out a very simple correspondence between these series and their p-parts.

To each reduced root system Φ of rank r, and each sufficiently large integer n, we define a family of multiple Dirichlet series in r complex variables, whose group of functional equations is isomorphic to the Weyl group of Φ. The coefficients in these Dirichlet series exhibit a multiplicativity that reduces the specification of the coefficients to those that are powers of a single prime p. For each p, the number of nonzero such coefficients is equal to the order of the Weyl group, and each nonzero coefficient is a product of n-th order Gauss sums. The root system plays a basic role in the combinatorics underlying the proof of the functional equations.

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.

The lower order and linear order of multiple dirichlet series

A Weyl group multiple Dirichlet seriesis a Dirichlet series in several complex variables attached to a root system Φ. The number of variables equals the rank rof the root system, and the series satisfies a group of functional equations isomorphic to the Weyl group Wof Φ. 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 that of [11, 10]; namely, we construct a rational function in rvariables invariant under a certain action of W, and use this to build a “local factor” of the global series.

In a recent paper (Proc. Roy. Soc. Edin., vol. lix, 1939, pp. 49–54) expressions in multiple series with argument 1 – z were found for Generalised Hypergeometric Functions of the typewhere p > 1. These formulæ are generalisations of known formulæ for the ordinary hypergeometric function F(α, β; ρ; z), and they are established by induction. The same method will be here employed to obtain generalisations of other known formulæ, these generalisations being in the form of multiple series.

It is shown, under some expected technical assumption, that a large class of multiple Dirichlet series which arise in the study of moments of L-functions have natural boundaries. As a remedy, we consider a new class of multiple Dirichlet series whose elements have nice properties: a functional equation and meromorphic continuation. This class suggests a notion of integral moments of L-functions.KeywordsMultiple Dirichlet seriesIntegral moments of L-functionsGood’s methodEisenstein seriesGL(3)

We describe surprising relationships between automorphic forms of various kinds, imaginary quadratic number fields and a certain system of six finite groups that are parameterised naturally by the divisors of twelve. The Mathieu group correspondence recently discovered by Eguchi-Ooguri-Tachikawa is recovered as a special case. We introduce a notion of extremal Jacobi form and prove that it characterises the Jacobi forms arising by establishing a connection to critical values of Dirichlet series attached to modular forms of weight two. These extremal Jacobi forms are closely related to certain vector-valued mock modular forms studied recently by Dabholkar-Murthy-Zagier in connection with the physics of quantum black holes in string theory. In a manner similar to monstrous moonshine the automorphic forms we identify constitute evidence for the existence of infinite-dimensional graded modules for the six groups in our system. We formulate an umbral moonshine conjecture that is in direct analogy with the monstrous moonshine conjecture of Conway-Norton. Curiously, we find a number of Ramanujan's mock theta functions appearing as McKay-Thompson series. A new feature not apparent in the monstrous case is a property which allows us to predict the fields of definition of certain homogeneous submodules for the groups involved. For four of the groups in our system we find analogues of both the classical McKay correspondence and McKay's monstrous Dynkin diagram observation manifesting simultaneously and compatibly.

Let \(\mu (n)\) be the Mobius function, \(\Lambda (n)\) be the von Mangoldt function, and \(A_F(n, 1,\ldots ,1)\) denote the nth coefficient of the Dirichlet series for L(s, F) associated to a Hecke–Maass form F for \(SL(m,\mathbb {Z})\). In this paper, as an appendix to our previous work, we firstly proved that for a Maass form for \(SL(3,\mathbb {Z})\), uniformly for all real numbers \(\theta \), the sequences \(\{\mu (n)\}\) and \(\{A_F(n,1)e(n\theta )\}\) are strong asymptotically orthogonal. Then as an analogue of Baker and Harman’s result on exponential sums formed with the Mobius function under the Generalized Riemann Hypothesis, we investigated the best possible estimates for the sum \(\sum _{n\le x}\Lambda (n) A_F(n, 1,\ldots ,1) e(n^k\theta )\) under certain assumptions.

On the description of an exceptional set in Borel's relation for multiple Dirichlet series with upper restriction on the growth

Dans cet article, nous considerons que certaines series de Dirichlet multiples, dont nous montrons le prolongement analytique en utilisant la formule integrale de Mellin-Barnes. Des majorations de ces series sont egalement obtenues.

The article deals with the nonlinear second order parabolic equation, which is known as heat equation with a source or the generalized porous medium equation. We construct solutions of a special type that describe disturbances propagating over a zero background with a finite velocity. Such solutions are called heat waves. Previously, we considered such problems in cases of one spatial variable, or without a source. In the present work, the obtained results are expanded to the case of two spatial variables. It leads to a transition to the polar coordinate system. The theorem on the existence and uniqueness of a solution, having the form of a heat wave, is proved for one boundary-value problem with special type degeneration in the class of analytical functions. The solution has the form of a multiple power series whose coefficients are determined when solving three-diagonal systems of linear algebraic equations with an infinitely increasing dimension. The convergence of the series is proved by the majorant method. We present an example where the conditions of the theorem are not fulfilled and show that the solution has the form of an immobile heat wave.

For quasi-neutral regions of semiconductor devices with position-dependent composition, we have derived expressions for the position dependence of the excess minority-carrier density and for relevant recombination currents. To make the development concrete, we study nonuniformly and heavily doped emitter regions of silicon p-n junction devices. The expressions developed differ from those previously advanced in that they are in the form of a multiple integral series, yielding, by truncation, many different orders of approximation. Correspondences exist between some of the different orders of approximation and various solutions previously obtained. All of the mechanisms relating to hole and electron transport in position-dependent heavily doped semiconductors are accounted for in the new expressions. These mechanisms include bandgap narrowing, majority-carrier degeneracy, Auger recombination lifetime, etc. To assess the accuracy of the various orders of approximation, we compare their predictions with a numerical solution. We determine that the simplest approximation, which contains only one term of the integral series, is accurate to within about 5 percent of the numerical solution for thin emitters (∼0.2 µm) provided the surface recombination velocity is less than 105cm/s. It thus applies directly to bipolar transistors with polysilicon contacts, and to surface-passivated solar cells. This new solution, which we call the zeroth-order or quasi-neutral quasi-equilibrium approximation, is simpler than solutions previously put forward. If the emitter junction is deep or the contact is ohmic, the higher-order approximations provide whatever accuracy is needed. We separate the emitter recombination current into charge and a characteristic time, enabling the calculation of the contribution of emitter to capacitance and to delay in the time domain or phase shift in the frequency domain.