Quadratic Dirichlet L-functions Research Articles

The fourth moment of quadratic Dirichlet L-functions over function fields

We obtain an asymptotic formula for the fourth moment of quadratic Dirichlet L-functions over $${\mathbb{F}_q[x]}$$ , as the base field $${\mathbb{F}_q}$$ is fixed and the genus of the family goes to infinity. According to conjectures of Andrade and Keating, we expect the fourth moment to be asymptotic to $${q^{2g+1} P(2g+1)}$$ up to an error of size $${o(q^{2g+1})}$$ , where P is a polynomial of degree 10 with explicit coefficients. We prove an asymptotic formula with the leading three terms, which agrees with the conjectured result.

Average values of L-series for real characters in function fields

We establish asymptotic formulae for the first and second moments of quadratic Dirichlet L-functions, at the center of the critical strip, associated to the real quadratic function field \(k(\sqrt{P})\) and inert imaginary quadratic function field \(k(\sqrt{\gamma P})\) with P being a monic irreducible polynomial over a fixed finite field \(\mathbb {F}_{q}\) of odd cardinality q and \(\gamma \) a generator of \(\mathbb {F}_{q}^{\times }\). We also study mean values for the class number and for the cardinality of the second K-group of maximal order of the associated fields for ramified imaginary, real, and inert imaginary quadratic function fields over \(\mathbb {F}_{q}\). One of the main novelties of this paper is that we compute the second moment of quadratic Dirichlet L-functions associated to monic irreducible polynomials. It is worth noting that the similar second moment over number fields is unknown. The second innovation of this paper comes from the fact that, if the cardinality of the ground field is even then the task of average L-functions in function fields is much harder and, in this paper, we are able to handle this strenuous case and establish several mean values results of L-functions over function fields.

The second and third moment of L(1/2,χ) in the hyperelliptic ensemble

Abstract We obtain asymptotic formulas for the second and third moment of quadratic Dirichlet L-functions at the critical point, in the function field setting. We fix the ground field 𝔽 q {\mathbb{F}_{q}} , and assume for simplicity that q is a prime with q ≡ 1 ⁢ ( mod ⁢ 4 ) {q\equiv 1~{}(\mathrm{mod}\,4)} . We compute the second and third moment of L ⁢ ( 1 / 2 , χ D ) {L(1/2,\chi_{D})} , when D is a monic square-free polynomial of degree 2 ⁢ g + 1 {2g+1} , as g → ∞ {g\to\infty} . The answer we get for the second moment agrees with Andrade and Keating’s conjectured formula in [4]. For the third moment, we check that the leading term agrees with the conjecture.

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

We investigate the first moment of the second derivative of quadratic Dirichlet L-functions over the rational function field. We establish an asymptotic formula when the cardinality of the finite field is fixed and the genus of the hyperelliptic curves associated to a family of Dirichlet L-functions over Fq(T) tends to infinity. As a more general result, we compute the full degree three polynomial in the asymptotic expansion of the first moment of the second derivative of this particular family of L-functions.

Rudnick and Soundararajan's theorem for function fields

In this paper we prove a function field version of a theorem by Rudnick and Soundararajan about lower bounds for moments of quadratic Dirichlet L-functions. We establish lower bounds for the moments of quadratic Dirichlet L-functions associated to hyperelliptic curves of genus g over a fixed finite field Fq in the large genus g limit.

Newman's conjecture in various settings

TextDe Bruijn and Newman introduced a deformation of the Riemann zeta function ζ(s), and found a real constant Λ which encodes the movement of the zeros of ζ(s) under the deformation. The Riemann hypothesis is equivalent to Λ≤0. Newman conjectured Λ≥0, remarking “the new conjecture is a quantitative version of the dictum that the Riemann hypothesis, if true, is only barely so.” Previous work could only handle ζ(s) and quadratic Dirichlet L-functions, obtaining lower bounds very close to zero (−1.14541⋅10−11 for ζ(s) and −1.17⋅10−7 for quadratic Dirichlet L-functions). We generalize to automorphic L-functions and function field L-functions, and explore the limit of these techniques. If D∈Z[T] is a square-free polynomial of degree 3 and Dp the polynomial in Fp[T] obtained by reducing D modulo p, we prove the Newman constant ΛDp equals log⁡|ap(D)|2p; by Sato–Tate (if the curve is non-CM) there exists a sequence of primes such that limn→∞⁡ΛDpn=0. We end by discussing connections with random matrix theory. VideoFor a video summary of this paper, please visit http://youtu.be/8A1XZtSkp_Q. This author video is a recording of a talk given by Alan Chang at CANT on May 28, 2014.

Low-lying Zeros of Quadratic Dirichlet L-Functions, Hyper-elliptic Curves and Random Matrix Theory

The statistics of low-lying zeros of quadratic Dirichlet L-functions were conjectured by Katz and Sarnak to be given by the scaling limit of eigenvalues from the unitary symplectic ensemble. The n-level densities were found to be in agreement with this in a certain neighborhood of the origin in the Fourier domain by Rubinstein in his Ph.D. thesis in 1998. An attempt to extend the neighborhood was made in the Ph.D. thesis of Peng Gao (n-level density of the low-lying zeros of quadratic Dirichlet L-functions, 2005), who under GRH gave the density as a complicated combinatorial factor, but it remained open whether it coincides with the Random Matrix Theory factor. For n ≤ 7 this was recently confirmed by Levinson and Miller. We resolve this problem for all n, not by directly doing the combinatorics, but by passing to a function field analogue, of L-functions associated to hyper-elliptic curves of given genus g over a field of q elements. We show that the answer in this case coincides with Gao’s combinatorial factor up to a controlled error. We then take the limit of large finite field size q → ∞ and use the Katz–Sarnak equidistribution theorem, which identifies the monodromy of the Frobenius conjugacy classes for the hyperelliptic ensemble with the group USp(2g). Further taking the limit of large genus g→ ∞ allows us to identify Gao’s combinatorial factor with the RMT answer.

Note on the mean value ofL(12,χ)in the hyperelliptic ensemble

In this paper we obtain an asymptotic formula for the first moment of quadratic Dirichlet L-functions L(s,χD) over the rational function field at the central point s=12, where D runs over all monic square-free polynomials of even degree.

N-Level Density of the Low-lying Zeros of Quadratic Dirichlet L-Functions

The Density Conjecture of Katz and Sarnak associates a classical compact group to each reasonable family of $L$-functions. Under the assumption of the Generalized Riemann Hypothesis, Rubinstein computed the $n$-level density of low-lying zeros for the family of quadratic Dirichlet $L$-functions in the case that the Fourier transform $\hat{f}(u)$ of any test function $f$ is supported in the region $\sum^n_{j=1}u_j < 1$ and showed that the result agrees with the Density Conjecture. In this paper, we improve Rubinstein's result on computing the $n$-level density for the Fourier transform $\hat{f}(u)$ being supported in the region $\sum^n_{j=1}u_j < 2$.

The n-level densities of low-lying zeros of quadratic Dirichlet L-functions

ABSTRACT. Previous work by Rubinstein [Rub] and Gao [Gao] computed the n-level densities for families of quadratic Dirichlet L-functions for test functions φ1, . . . , φn supported in ∑ n i=1 |ui| < 2, and showed agreement with random matrix theory predictions in this range for n ≤ 3 but only in a restricted range for larger n. We extend these results and show agreement for n ≤ 7, and reduce higher n to a Fourier transform identity. The proof involves adopting a new combinatorial perspective to convert all terms to a canonical form, which facilitates the comparison of the two sides.

Lower order terms for the moments of symplectic and orthogonal families of L -functions

We derive formulas for the terms in the conjectured asymptotic expansions of the moments, at the central point, of quadratic Dirichlet L -functions, L ( 1 / 2 , χ d ) , and also of the L -functions associated to quadratic twists of an elliptic curve over Q . In so doing, we are led to study determinants of binomial coefficients of the form det ( ( 2 k − i − λ k − i + 1 2 k − 2 j ) ) .

The third moment of quadratic Dirichlet L-functions

We study the third moment of quadratic Dirichlet \(L\)-functions, obtaining an error term of size \(O(X^{3/4 + \varepsilon })\).

Conjectures and Experiments Concerning the Moments of L(1/2, χ d )

We report on some extensive computations and experiments concerning the moments of quadratic Dirichlet L-functions at the critical point. We computed the values of L(1/2, χ d ) for −5×1010<d<1.3×1010 in order to numerically test conjectures concerning the moments ∑|d|<X L(1/2, χ d ) k . Specifically, we tested the full asymptotics for the moments conjectured by Conrey, Farmer, Keating, Rubinstein, and Snaith, as well as the conjectures of Diaconu, Goldfeld, Hoffstein, and Zhang concerning additional lower-order terms in the moments. We also describe the algorithms used for this large-scale computation.

The mean value ofL(12,χ)in the hyperelliptic ensemble

We obtain an asymptotic formula for the first moment of quadratic Dirichlet L-functions over the rational function field at the central point s=12. Specifically, we compute the expected value of L(12,χ) for an ensemble of hyperelliptic curves of genus g over a fixed finite field as g→∞. Our approach relies on the use of the analogue of the approximate functional equation for such L-functions. The results presented here are the function field analogues of those obtained previously by Jutila in the number-field setting and are consistent with recent general conjectures for the moments of L-functions motivated by Random Matrix Theory.

On the Distribution of Extreme Values of Zeta and L-Functions in the Strip 1/2 &lt; &lt; 1

Abstract. We study the distribution of large (and small) values of several families of L-functions on a line Re(s) = σ where 1/2 < σ < 1. We consider the Riemann zeta function ζ(s) in the t-aspect, Dirichlet L-functions in the q-aspect, and L-functions attached to primitive holomorphic cusp forms of weight 2 in the level aspect. For each family we show that the L-values can be very well modeled by an adequate random Euler product, uniformly in a wide range. We also prove new Ω-results for quadratic Dirichlet L-functions (predicted to be best possible by the probabilistic model) conditionally on GRH, and other results related to large moments of ζ(σ+ it).

Moments of Quadratic Dirichlet L-Functions over Rational Function Fields

Moments of Quadratic Dirichlet L-Functions over Rational Function Fields

The first moment of quadratic Dirichlet L-functions

We obtain an asymptotic formula for the smoothly weighted first moment of primitive quadratic Dirichlet L-functions at the central point, with an error term that is "square-root" of the main term. Our approach uses a recursive technique that feeds the result back into itself, successively improving the error term.

Symmetry Transitions in Random Matrix Theory &amp; L-functions

We compute the moments of the characteristic polynomials of random orthogonal and symplectic matrices, defined by averages with respect to Haar measure on SO(2N) and USp(2N), to leading order as N → ∞, on the unit circle as functions of the angle θ measured from one of the two symmetry points in the eigenvalue spectrum \({\{\exp(\pm i \theta_n)\}_{1\le n \le N}}\). Our results extend previous formulae that relate just to the symmetry points, i.e. to θ = 0. Local spectral statistics are expected to converge to those of random unitary matrices in the limit as N → ∞ when θ is fixed, and to show a transition from the orthogonal or symplectic to the unitary forms on the scale of the mean eigenvalue spacing: if θ = πy/N they become functions of y in the limit when N → ∞. We verify that this is true for the spectral two-point correlation function, but show that it is not true for the moments of the characteristic polynomials, for which the leading order asymptotic approximation is a function of θ rather than y. Symmetry points therefore influence the moments asymptotically far away on the scale of the mean eigenvalue spacing. We also investigate the moments of the logarithms of the characteristic polynomials in the same context. The moments of the characteristic polynomials of random matrices are conjectured to be related to the moments of families of L-functions. Previously, moments at the symmetry point θ = 0 have been related to the moments of families of L-functions evaluated at the centre of the critical strip. Our results motivate general conjectures for the moments of orthogonal and symplectic families of L-functions evaluated at a fixed height t up the critical line. These conjectures suggest that the symmetry of the non-trivial zeros of the L-functions influences the moments asymptotically far, on the scale of the mean zero spacing, from the centre of the critical strip. We verify that the second moments of real quadratic Dirichlet L-functions and a family of automorphic L-functions are consistent with our conjectures.

On the mean values of L -functions in orthogonal and symplectic families

Hybrid Euler–Hadamard products have previously been studied for the Riemann zeta function on its critical line and for Dirichlet L-functions, in the context of the calculation of moments and connections with Random Matrix Theory. According to the Katz–Sarnak classification, these are believed to represent families of L-function with unitary symmetry. We here extend the formalism to families with orthogonal and symplectic symmetry. Specifically, we establish formulae for real quadratic Dirichlet L-functions and for the L-functions associated with primitive Hecke eigenforms of weight 2 in terms of partial Euler and Hadamard products. We then prove asymptotic formulae for some moments of these partial products and make general conjectures based on results for the moments of characteristic polynomials of random matrices.

On the One-Level Density Conjecture for Quadratic Dirichlet L-Functions

AbstractIn a previous article, we studied the distribution of “low–lying” zeros of the family of quadratic Dirichlet L–functions assuming the Generalized Riemann Hypothesis for all Dirichlet L–functions. Even with this very strong assumption, we were limited to using weight functions whose Fourier transforms are supported in the interval (–2, 2). However, it is widely believed that this restriction may be removed, and this leads to what has become known as the One-Level Density Conjecture for the zeros of this family of quadratic L-functions. In this note, wemake use of Weil's explicit formula as modified by Besenfelder to prove an analogue of this conjecture.