Quadratic Dirichlet L-functions Research Articles

A NEW PROGRAM FOR THE ENTIRE FUNCTIONS IN NUMBER THEORY

In this paper, we propose a new program for introducing the sign of the functional equation to present the entire functions of order one in number theory. We suggest some open problems for the zeros of these entire functions related to the completed Dedekind zeta function, completed quadratic Dirichlet L-functions, completed Ramanujan zeta function and completed automorphic L-function. These lead to the contribution to giving the deep understanding for diffusion equations associated with the entire functions of order one in number theory.

Secondary terms in the asymptotics of moments of L-functions

We propose a refined version of the existing conjectural asymptotic formula for the moments of the family of quadratic Dirichlet L-functions over rational function fields. Our prediction is motivated by two natural conjectures that provide sufficient information to determine the analytic properties (meromorphic continuation, location of poles, and the residue at each pole) of a certain generating function of moments of quadratic L-functions. The number field analogue of our asymptotic formula can be obtained by a similar procedure, the only difference being the contributions coming from the archimedean and even places, which require a separate analysis. To avoid this additional technical issue, we present, for simplicity, the asymptotic formula only in the rational function field setting. This has also the advantage of being much easier to test.

Simultaneous non-vanishing of quadratic Dirichlet L-functions and twists of Hecke L-functions

Simultaneous non-vanishing of quadratic Dirichlet L-functions and twists of Hecke L-functions

Moments of quadratic Dirichlet L-functions over function fields

In this paper, we establish the expected order of magnitude of the kth-moment of quadratic Dirichlet L-functions associated to hyperelliptic curves of genus g as well as of prime moduli over a fixed finite field Fq for all real k≥0.

The Shintani double zeta functions

Abstract In this paper, we give an explicit formula for the Shintani double zeta functions with any ramification in the most general setting of adeles over an arbitrary number field. Three applications of the explicit formula are given. First, we obtain a functional equation satisfied by the Shintani double zeta functions in addition to Shintani’s functional equations. Second, we establish the holomorphicity of a certain Dirichlet series generalizing a result by Ibukiyama and Saito. This Dirichlet series occurs in the study of unipotent contributions of the geometric side of the Arthur–Selberg trace formula of the symplectic group. Third, we prove an asymptotic formula of the weighted average of the central values of quadratic Dirichlet L-functions.

The mixed second moment of quadratic Dirichlet L-functions over function fields

We investigate the mixed second moment involving the second derivative at the central point of a family of quadratic Dirichlet L-functions over 𝔽q(t), associated to the hyperelliptic curves of genus g. We compute the full degree five polynomial in the asymptotic expansion of the mixed second moment when the cardinality q of the finite field is fixed and the genus g tends to infinity. This is a partial analogue of classical Ingham’s result about the Riemann zeta function.

Second moment of the Prime Geodesic Theorem for $$\mathrm {PSL}(2, \mathbb {Z}[i])$$

The remainder \(E_\Gamma (X)\) in the Prime Geodesic Theorem for the Picard group \(\Gamma = \mathrm {PSL}(2,\mathbb {Z}[i])\) is known to be bounded by \(O(X^{3/2+\epsilon })\) under the assumption of the Lindelöf hypothesis for quadratic Dirichlet L-functions over Gaussian integers. By studying the second moment of \(E_\Gamma (X)\), we show that on average the same bound holds unconditionally.

The integral moments and ratios of quadratic Dirichlet L-functions over monic irreducible polynomials in mathbb {F}_{q}[T

In this paper, we extend to the function field setting the heuristics formerly developed by Conrey, Farmer, Keating, Rubinstein and Snaith, for the integral moments of L-functions. We also adapt to the function field setting the heuristics first developed by Conrey, Farmer and Zirnbauer to the study of mean values of ratios of L-functions. Specifically, the focus of this paper is on the family of quadratic Dirichlet L-functions L(s,chi _{P}) where the character chi is defined by the Legendre symbol for polynomials in mathbb {F}_{q}[T] with mathbb {F}_{q} a finite field of odd cardinality, and the averages are taken over all monic and irreducible polynomials P of a given odd degree. As an application, we also compute the formula for the one-level density for the zeros of these L-functions.

Lower order terms for the one-level density of a symplectic family of Hecke L-functions

In this paper we apply the L-function Ratios Conjecture to compute the one-level density for a symplectic family of L-functions attached to Hecke characters of infinite order. When the support of the Fourier transform of the corresponding test function f reaches 1, we observe a transition in the main term, as well as in the lower order term. The transition in the lower order term is in line with behavior recently observed by D. Fiorilli, J. Parks, and A. Södergren in their study of a symplectic family of quadratic Dirichlet L-functions [11,12]. We then directly calculate main and lower order terms for test functions f such that supp(fˆ)⊂[−α,α] for some α<1, and observe that this unconditional result is in agreement with the prediction provided by the Ratios Conjecture. As the analytic conductor of these L-functions grow twice as large (on a logarithmic scale) as the cardinality of the family in question, this is the optimal support that can be expected with current methods. Finally, as a corollary, we deduce that, under GRH, at least 75% of these L-functions do not vanish at the central point.

The fourth moment of quadratic Dirichlet L-functions

We study the fourth moment of quadratic Dirichlet L-functions at $$s= \frac{1}{2}$$ . We show an asymptotic formula under the generalized Riemann hypothesis, and obtain a precise lower bound unconditionally. The proofs of these results follow closely arguments of Soundararajan and Young (J Eur Math Soc 12(5):1097–1116, 2010) and Soundararajan (Ann Math (2) 152(2):447–488, 2000).

Prime geodesic theorem for the Picard manifold

Let Γ=PSL(2,Z[i]) be the Picard group and H3 be the three-dimensional hyperbolic space. We study the Prime Geodesic Theorem for the quotient Γ∖H3, called the Picard manifold, obtaining an error term of size O(X3/2+θ/2+ϵ), where θ denotes a subconvexity exponent for quadratic Dirichlet L-functions defined over Gaussian integers.

Weighted first moments of some special quadratic Dirichlet L-functions

In this paper, we obtain asymptotic formulas for weighted first moments of central values of families of primitive quadratic Dirichlet L-functions whose conductors comprise only primes that split in a given quadratic number field. We then deduce a non-vanishing result of these L-functions at the point s = 1 / 2.

The second moment of quadratic Dirichlet L-functions

In this paper, we compute the second moment of primitive quadratic Dirichlet L-functions with weight at the central point. By applying Young's methods ([11], [12]), we obtain an asymptotic formula for the second moment with an error term that is“square-root” of the main term.

Note on the mean values of derivatives of quadratic Dirichlet L-functions in function fields

We study the mean values of the first and the second derivative of quadratic Dirichlet L-functions L(s,χD) over the rational function field. We show that the moments of first derivatives L′(12,χD) are just constant multiples of the moments of L(12,χD). For the second derivatives, we improve the error term by q12(1+ϵ) and show that there is an extra term of size g3q2g+13 in the asymptotic formula of Andrade and Rajagopal for the first moment of L″(12,χD).

Mean values of derivatives of L-functions in function fields: III

AbstractIn this series of papers, we explore moments of derivatives of L-functions in function fields using classical analytic techniques such as character sums and approximate functional equation. The present paper is concerned with the study of mean values of derivatives of quadratic Dirichlet L-functions over function fields when the average is taken over monic and irreducible polynomials P in 𝔽q[T]. When the cardinality q of the ground field is fixed and the degree of P gets large, we obtain asymptotic formulas for the first moment of the first and the second derivative of this family of L-functions at the critical point. We also compute the full polynomial expansion in the asymptotic formulas for both mean values.

On the third moment of [formula omitted] I: The rational function field case

In this note, we prove the existence of a secondary term in the asymptotic formula of the cubic moment of quadratic Dirichlet L-functions∑d0−monic&sq.freedeg⁡d0=DL(12,χd0)3 over rational function fields on the order of q34D. This term is in perfect analogy with the x34-term indicated in our joint work [18] for the corresponding asymptotic formula over the rationals.

Vanishing of hyperelliptic L-functions at the central point

We obtain a lower bound on the number of quadratic Dirichlet L-functions over the rational function field which vanish at the central point s=1/2. This is in contrast with the situation over the rational numbers, where a conjecture of Chowla predicts there should be no such L-functions. The approach is based on the observation that vanishing at the central point can be interpreted geometrically, as the existence of a map to a fixed abelian variety from the hyperelliptic curve associated to the character.

Low-lying zeros of quadratic Dirichlet L-functions: A transition in the ratios conjecture

We study the 1-level density of low-lying zeros of quadratic Dirichlet L-functions by applying the L-functions Ratios Conjecture. We observe a transition in the main term as was predicted by the Ka ...

Mean value of the class number in function fields revisited

In this paper an asymptotic formula for the sum sum L(1,chi ) is established for the family of quadratic Dirichlet L-functions over the rational function field over a finite field mathbb {F}_{q} with q fixed. Using the recent techniques developed by Florea we obtain an extra lower order terms that was never been predicted in number fields and function fields. As a corollary, we obtain a formula for the average of the class number over function fields which also contains strenuous lower order terms and so improving on previous results of Hoffstein and Rosen.

A fast amortized algorithm for computing quadratic Dirichlet L-functions

An algorithm to compute Dirichlet $L$-functions for many quadratic characters is derived. The algorithm is optimal (up to logarithmic factors) provided that the conductors of the characters under consideration span a dyadic window.