Inversion formula for Dirichlet series and its application

We develop the theory of Weyl group multiple Dirichlet series for root systems of type C. For a 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. They conjecturally arise as Whittaker coefficients of Eisenstein series on a metaplectic group with cover degree n. For type C and n odd, we construct an infinite family of Dirichlet series and prove they satisfy the above analytic properties in many cases. The coefficients are exponential sums built from Gelfand‐Tsetlin bases of certain highest weight representations. Previous attempts to define such series by Brubaker, Bump, and Friedberg required n sufficiently large, so that coefficients were described by Weyl group orbits. We demonstrate that these two radically different descriptions match when both are defined. Moreover, for n D 1, we prove our series are Whittaker coefficients of Eisenstein series on SO.2r C 1/.

The object of this paper is to prove that these theorems are true even when r > 1. The argument used is similar to that used in Theorem 22 of the Cambridge Tract on The General Theory of Dirichlet's Series by Hardy and Riesz. Some remarks are also made in the concluding section about the applications of these Tauberian theorems to obtain certain precise results on the abcissae of summability of Dirichlet's series. These results have been anticipated by Ananda Rau.

In this note, we prove an inversion formula for Dirichlet polynomials involving a non-principal character. By using Heath-Brown's upper-bound for Dirichlet's L-functions, we are able to obtain a non-trivial upper-bound for the error term in the q aspect. As an application we obtain an approximate functional equation for Dirichlet's L-functions to the right of the critical line with a sharp error term.

Kubota [T. Kubota, Some results concerning reciprocity law and real analytic automorphic functions, in: 1969 Number Theory Institute (Proc. Sympos. Pure Math. XX, State Univ. New York, Stony Brook, N.Y., 1969), Amer. Math. Soc., Providence, R.I. (1971), 382–395.] showed how the theory of Eisenstein series on the higher metaplectic covers of SL2 (which he discovered) can be used to study the analytic properties of Dirichlet series formed with n-th order Gauss sums. In this paper we will prove a functional equation for such Dirichlet series in the precise form required by the companion paper [B. Brubaker, D. Bump, G. Chinta, S. Friedberg, and J. Hostein, Weyl group multiple Dirichlet series I, preprint, http://sporadic.stanford.edu/bump/wmd.pdf.]. Closely related results are in Eckhardt and Patterson [C. Eckhardt and S. J. Patterson, On the Fourier coefficients of biquadratic theta series, Proc. London Math. Soc. (3) 64(2) (1992), 225–264.].

In this paper, we prove multiple analogues of famous Ramanujan’s formulas for certain Dirichlet series which were introduced in his well-known notebooks. Furthermore, we prove some multiple versions of analogous formulas of Ramanujan which were given by Berndt and so on.

Under the generalized Lindelof 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.

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.

On Dirichlet series and functional equations

We derive a Lorentzian OPE inversion formula for the principal series of sl(2, ℝ). Unlike the standard Lorentzian inversion formula in higher dimensions, the formula described here only applies to fully crossing-symmetric four-point functions and makes crossing symmetry manifest. In particular, inverting a single conformal block in the crossed channel returns the coefficient function of the crossing-symmetric sum of Witten exchange diagrams in AdS, including the direct-channel exchange. The inversion kernel exhibits poles at the double-trace scaling dimensions, whose contributions must cancel out in a generic solution to crossing. In this way the inversion formula leads to a derivation of the Polyakov bootstrap for sl(2, ℝ). The residues of the inversion kernel at the double-trace dimensions give rise to analytic bootstrap functionals discussed in recent literature, thus providing an alternative explanation for their existence. We also use the formula to give a general proof that the coefficient function of the principal series is meromorphic in the entire complex plane with poles only at the expected locations.

Converse Theorems traditionally have provided a way to characterize Dirichlet series associated to modular forms in terms of their analytic properties. The prototypical Converse Theorem was due to Hamburger who characterized the Riemann zeta function in terms of its analytic properties [23]. More familiar may be the Converse Theorems of Hecke and Weil. Hecke first proved that Dirichlet series associated to modular forms enjoyed “nice” analytic properties and then proved “Conversely” that these analytic properties in fact characterized modular Dirichlet series [26]. Weil extended this Converse Theorem to Dirichlet series associated to modular forms with level [62]. In their modern formulation, Converse Theorems are stated in terms of automorphic representations instead of modular forms. This was first done by Jacquet and Langlands for GL2 [31]. For GLn, Jacquet, Piatetski-Shapiro, and Shalika have proved that the L-functions associated to automorphic representations have nice analytic properties similar to those of Hecke [32] (see also [15, 9]). The relevant “nice” properties are: analytic continuation, boundedness in vertical strips, and functional equation. Converse Theorems in this context invert this process and give a criterion for automorphy, or modularity, in terms of these L-functions being “nice” [31, 32, 13, 9]. The first application of a Converse Theorem that might come to mind is to the question of modularity of arithmetic or geometric objects. To use the Converse Theorem in this context one must first be able to prove that the Lfunctions of these arithmetic/geometric objects are “nice”. However, essentially the only way to show that an L-function is nice is to have it associated to an automorphic form already! So, on second thought, as a direct tool for establishing modularity the Converse Theorem seems a bit lacking. Instead, the most natural direct application of the Converse Theorem is to Functoriality, in this case the

The Riesz means, or sometimes typical means, were introduced by M. Riesz and have been studied in connection with summability of Fourier series and of Dirichlet series [8] and [11]. In number-theoretic context, it is the Riesz sum rather than the Riesz mean that has been extensively studied. The Riesz sums appear as long as there appears the G-function. Cf. Remark 1 and [14]. As is shown below, the Riesz sum corresponds to integration while Landau's differencing is an analogue of differentiation. This integration-differentiation aspect has been the driving force of many researches on number-theoretic asymptotic formulas. Ingham's decent treatment [13] of the prime number theorem is one typical example. We state some efficient theorems that give asymptotic formulas for the sums of coefficients of the generating Dirichlet series not necessarily satisfying the functional equation.

We solve Maxwell's eigenvalue problem via isogeometric boundary elements and a contour integral method. We discuss the analytic properties of the discretization, outline the implementation, and showcase numerical examples.

This article gives the definition of interval-valued Dirichlet series. Based on [1, , this article discusses some properties about the coefficient of interval-valued and fuzzy Dirichlet series. Ihe calculation of coefficient , of interval-valued Dirichlet series and the coefficient of fuzzy Dirichelt series are also discussed. Moreover, some relative theorems and properties are presented.

Weyl group multiple Dirichlet series are generalizations of the Riemann zeta function. Like the Riemann zeta function, they are Dirichlet series with analytic continuation and functional equations, having applications to analytic number theory. By contrast, these Weyl group multiple Dirichlet series may be functions of several complex variables and their groups of functional equations may be arbitrary finite Weyl groups. Furthermore, their coefficients are multiplicative up to roots of unity, generalizing the notion of Euler products. This book proves foundational results about these series and develops their combinatorics. These interesting functions may be described as Whittaker coefficients of Eisenstein series on metaplectic groups, but this characterization doesn't readily lead to an explicit description of the coefficients. The coefficients may be expressed as sums over Kashiwara's crystals, which are combinatorial analogs of characters of irreducible representations of Lie groups. For Cartan Type A, there are two distinguished descriptions, and if these are known to be equal, the analytic properties of the Dirichlet series follow. Proving the equality of the two combinatorial definitions of the Weyl group multiple Dirichlet series requires the comparison of two sums of products of Gauss sums over lattice points in polytopes. Through a series of surprising combinatorial reductions, this is accomplished. The book includes expository material about crystals, deformations of the Weyl character formula, and the Yang–Baxter equation.

We discuss how one could study asymptotics of cyclotomic quantities via the mean values of certain multiplicative functions and their Dirichlet series using a theorem of Delange. We show how this could provide a new approach to Artin's conjecture on primitive roots. We focus on whether a fixed prime has a certain order modulo infinitely many other primes. We also give an estimate for the mean value of one such Dirichlet series.