Quadratic Dirichlet Research Articles

On the Paley graph of a quadratic character

Abstract Paley graphs form a nice link between the distribution of quadratic residues and graph theory. These graphs possess remarkable properties which make them useful in several branches of mathematics. Classically, for each prime number p we can construct the corresponding Paley graph using quadratic and non-quadratic residues modulo p. Therefore, Paley graphs are naturally associated with the Legendre symbol at p which is a quadratic Dirichlet character of conductor p. In this article, we introduce the generalized Paley graphs. These are graphs that are associated with a general quadratic Dirichlet character. We will then provide some of their basic properties. In particular, we describe their spectrum explicitly. We then use those generalized Paley graphs to construct some new families of Ramanujan graphs. Finally, using special values of L-functions, we provide an effective upper bound for their Cheeger number. As a by-product of our approach, we settle a question raised in [Cramer et al.: The isoperimetric and Kazhdan constants associated to a Paley graph, Involve 9 (2016), 293–306] about the size of this upper bound.

BOUNDS FOR MOMENTS OF QUADRATIC DIRICHLET CHARACTER SUMS

BOUNDS FOR MOMENTS OF QUADRATIC DIRICHLET CHARACTER SUMS

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.

Negative Moments of L–Functions With Small Shifts Over Function Fields

Abstract We consider negative moments of quadratic Dirichlet $L$–functions over function fields. Summing over monic square-free polynomials of degree $2g+1$ in $\mathbb{F}_{q}[x]$, we obtain an asymptotic formula for the $k^{\textrm{th}}$ shifted negative moment of $L(1/2+\beta ,\chi _{D})$, in certain ranges of $\beta $ (e.g., when roughly $\beta \gg \log g/g $ and $k<1$). We also obtain non-trivial upper bounds for the $k^{\textrm{th}}$ shifted negative moment when $\log (1/\beta ) \ll \log g$. Previously, almost sharp upper bounds were obtained in [ 3] in the range $\beta \gg g^{-\frac{1}{2k}+\epsilon }$.

Numerical estimates on the Landau-Siegel zero and other related quantities

Let q be an odd prime, χ be a non-principal Dirichlet character modq and L(s,χ) be the associated Dirichlet L-function. For every odd prime q≤107, we show that L(1,χ□)>c1log⁡q and β<1−c2log⁡q, where c1=0.0124862668…, c2=0.0091904477…, χ□ is the quadratic Dirichlet character modq and β∈(0,1) is the Landau-Siegel zero, if it exists, of the set of such Dirichlet L-functions. As a by-product of the computations here performed, we also obtained some information about Littlewood's and Joshi's bounds on L(1,χ□) and on the class number of the imaginary quadratic field Q(−q).

Jacobi forms, Saito-Kurokawa lifts, their Pullbacks and sup-norms on average

We formulate a precise conjecture about the size of the $$L^\infty $$ -mass of the space of Jacobi forms on $$\mathbb {H}_n \times \mathbb C^{g \times n}$$ of matrix index S of size g. This $$L^\infty $$ -mass is measured by the size of the Bergman kernel of the space. We prove the conjectured lower bound for all such n, g, S and prove the upper bound in the k aspect when $$n=1$$ , $$g \ge 1$$ . When $$n=1$$ and $$g=1$$ , we make a more refined study of the sizes of the index-(old and) new spaces, the latter via the Waldspurger’s formula. Towards this and with independent interest, we prove a power saving asymptotic formula for the averages of the twisted central L-values $$L(1/2, f \otimes \chi _D)$$ with f varying over newforms of level a prime p and even weight k as $$k,p \rightarrow \infty $$ and D being (explicitly) polynomially bounded by k, p. Here $$\chi _D$$ is a real quadratic Dirichlet character. We also prove that the size of the space of Saito-Kurokawa lifts (of even weight k) is $$k^{5/2}$$ by three different methods (with or without the use of central L-values), and show that the size of their pullbacks to the diagonally embedded $$\mathbb {H}\times \mathbb {H}$$ is $$k^2$$ . In an appendix, the same question is answered for the pullbacks of the whole space $$S^2_k$$ , the size here being $$k^3$$ .

Bounds for moments of quadratic Dirichlet $L$-functions of prime-related moduli

In this paper, we study the $k$-th moment of central values of the family of quadratic Dirichlet $L$-functions of moduli $8p$, with $p$ ranging over odd primes. Assuming the truth of the generlized Riemann hypothesis, we establish sharp upper and lower bounds for the $k$-th power moment of these $L$-values for all real $k \geq 0$.

One-level density of quadratic twists of $L$-functions

In this paper, we investigate the one-level density of low-lying zeros of quadratic twists of automorphic $L$-functions under the generalized Riemann hypothesis and the Ramanujan-Petersson conjecture. We improve upon the known results using only functional equations for quadratic Dirichlet $L$-functions.

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

The prime geodesic theorem for PSL2(ℤ[i]) and spectral exponential sums

This work addresses the Prime Geodesic Theorem for the Picard manifold $\mathcal{M} = \mathrm{PSL}_{2}(\mathbb{Z}[i]) \backslash \mathfrak{h}^{3}$, which asks for the asymptotic evaluation of a counting function for the closed geodesics on $\mathcal{M}$. Let $E_{\Gamma}(X)$ be the error term in the Prime Geodesic Theorem. We establish that $E_{\Gamma}(X) = O_{\varepsilon}(X^{3/2+\varepsilon})$ on average as well as many pointwise bounds. The second moment bound parallels an analogous result for $\Gamma = \mathrm{PSL}_{2}(\mathbb{Z})$ due to Balog et al. and our innovation features the delicate analysis of sums of Kloosterman sums with an explicit manipulation of oscillatory integrals. The proof of the pointwise bounds requires Weyl-strength subconvexity for quadratic Dirichlet $L$-functions over $\mathbb{Q}(i)$. Moreover, an asymptotic formula for a spectral exponential sum in the spectral aspect for a cofinite Kleinian group $\Gamma$ is given. Our numerical experiments visualise in particular that $E_{\Gamma}(X)$ obeys a conjectural bound of size $O_{\epsilon}(X^{1+\varepsilon})$.

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.

One-level density of the family of twists of an elliptic curve over function fields

We fix an elliptic curve E/Fq(t) and consider the family {E⊗χD} of E twisted by quadratic Dirichlet characters. The one-level density of their L-functions is shown to follow orthogonal symmetry for test functions with Fourier transform supported inside (−1,1). As an application, we obtain an upper bound of 3/2 on the average analytic rank. By splitting the family according to the sign of the functional equation, we obtain that at least 12.5% of the family have rank zero, and at least 37.5% have rank one. The Katz and Sarnak philosophy predicts that those percentages should both be 50% and that the average analytic rank should be 1/2. We finish by computing the one-level density of E twisted by Dirichlet characters of order ℓ≠2 coprime to q. We obtain a restriction of (−1/2,1/2) on the support with a unitary symmetry.

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.

Solvable lattice sums and quadratic Dirichlet $L$-values

We study how to express the Epstein zeta functions of binary quadratic forms in terms of a finite number of quadratic Dirichlet $L$-values. We prove a conjecture raised by Zucker and Robertson in 1984.

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.

A short note on inadmissible coefficients of weight 2 and 2k+1 newforms

Let f(z)=q+sum _{nge 2}a(n)q^n be a weight k normalized newform with integer coefficients and trivial residual mod 2 Galois representation. We extend the results of Amir and Hong in Amir and Hong (On L-functions of modular elliptic curves and certain K3 surfaces, Ramanujan J, 2021) for k=2 by ruling out or locating all odd prime values |ell |<100 of their Fourier coefficients a(n) when n satisfies some congruences. We also study the case of odd weights kge 1 newforms where the nebentypus is given by a quadratic Dirichlet character.

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.

On the third moment of $L(\frac{1}{2}, \chi_{d})$ {II}: the number field case

We establish a smoothed asymptotic formula for the third moment of quadratic Dirichlet L -functions at the central value. In addition to the main term, which is known, we prove the existence of a secondary term of size x^{\frac{3}{4}}. The error term in the asymptotic formula is on the order of O(x^{\frac{2}{3}+\delta}) for every \delta &gt; 0.