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

Jake Levinson; Steven J. Miller

Displaying similar documents to “The n-level densities of low-lying zeros of quadratic Dirichlet L-functions”

On zeros of Dirichlet’s $L$ -functions.

Н.Г. Чудаков ([unknown])

Matematiceskij sbornik

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Admissible functions for the Dirichlet space

Javad Mashreghi, Mahmood Shabankhah (2010)

Studia Mathematica

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Zero sets and uniqueness sets of the classical Dirichlet space are not completely characterized yet. We define the concept of admissible functions for the Dirichlet space and then apply them to obtain a new class of zero sets for . Then we discuss the relation between the zero sets of and those of $^{\infty}$ .

On Dirichlet type spaces on the unit ball of $ℂ^{n}$

Małgorzata Michalska (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In this paper we discuss characterizations of Dirichlet type spaces on the unit ball of $ℂ^{n}$ obtained by P. Hu and W. Zhang [2], and S. Li [4].

On an entire function represented by multiple Dirichlet series

Lakshika Chutani (2021)

Mathematica Bohemica

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Consider the space $L$ of entire functions represented by multiple Dirichlet series that becomes a non uniformly convex Banach space which is also proved to be dense, countable and separable. Continuing further, for the given space $L$ the characterization of bounded linear transformations in terms of matrix and characterization of linear functional has been obtained.

The Dirichlet-Bohr radius

Daniel Carando, Andreas Defant, Domingo A. Garcí, Manuel Maestre, Pablo Sevilla-Peris (2015)

Acta Arithmetica

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Denote by Ω(n) the number of prime divisors of n ∈ ℕ (counted with multiplicities). For x∈ ℕ define the Dirichlet-Bohr radius L(x) to be the best r > 0 such that for every finite Dirichlet polynomial $\sum_{n \leq x} a_{n} n^{- s}$ we have $\sum_{n \leq x} | a_{n} | r^{Ω (n)} \leq s u p_{t \in ℝ} | \sum_{n \leq x} a_{n} n^{- i t} |$ . We prove that the asymptotically correct order of L(x) is ${(l o g x)}^{1 / 4} x^{- 1 / 8}$ . Following Bohr’s vision our proof links the estimation of L(x) with classical Bohr radii for holomorphic functions in several variables. Moreover, we suggest a general setting which allows translating various results...

On the Dirichlet problem associated with the Dunkl Laplacian

Mohamed Ben Chrouda (2016)

Annales Polonici Mathematici

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This paper deals with the questions of the existence and uniqueness of a solution to the Dirichlet problem associated with the Dunkl Laplacian $Δ_{k}$ as well as the hypoellipticity of $Δ_{k}$ on noninvariant open sets.

Dirichlet forms on quotients of shift spaces

Manfred Denker, Atsushi Imai, Susanne Koch (2007)

Colloquium Mathematicae

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We define thin equivalence relations ∼ on shift spaces $^{\infty}$ and derive Dirichlet forms on the quotient space $Σ =^{\infty} / \sim$ in terms of the nearest neighbour averaging operator. We identify the associated Laplace operator. The conditions are applied to some non-self-similar extensions of the Sierpiński gasket.

On unitary equivalence of invariant subspaces of the Dirichlet space

Kunyu Guo, Liankuo Zhao (2010)

Studia Mathematica

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It is shown that in the Dirichlet space , two invariant subspaces ℳ ₁, ℳ ₂ of the Dirichlet shift $M_{z}$ are unitarily equivalent only if ℳ ₁ = ℳ ₂.

Envelope of Dirichlet problems on a domain in $C^{n}$

Eric Bedford (1985)

Annales Polonici Mathematici

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An asymptotic formula related to the divisors of the quaternary quadratic form

Liqun Hu (2014)

Acta Arithmetica

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Let d(n) stand for the Dirichlet divisor function. We give an asymptotic formula for $\sum_{1 \leq m ₁, m ₂, m ₃, m ₄ \leq x} d (m ₁ ² + m ₂ ² + m ₃ ² + m ₄ ²)$ with the help of the circle method.

The Bohr inequality for ordinary Dirichlet series

R. Balasubramanian, B. Calado, H. Queffélec (2006)

Studia Mathematica

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We extend to the setting of Dirichlet series previous results of H. Bohr for Taylor series in one variable, themselves generalized by V. I. Paulsen, G. Popescu and D. Singh or extended to several variables by L. Aizenberg, R. P. Boas and D. Khavinson. We show in particular that, if $f (s) = \sum_{n = 1}^{\infty} a ₙ n^{- s}$ with ${| | f | |}_{\infty} : = s u p_{ℜ s > 0} | f (s) | < \infty$ , then $\sum_{n = 1}^{\infty} | a ₙ | n^{- 2} {\leq | | f | |}_{\infty}$ and even slightly better, and $\sum_{n = 1}^{\infty} | a ₙ | n^{- 1 / 2} {\leq C | | f | |}_{\infty}$ , C being an absolute constant.

On the Dirichlet problem in the Cegrell classes

Rafał Czyż, Per Åhag (2004)

Annales Polonici Mathematici

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Let μ be a non-negative measure with finite mass given by $φ (d d^{c} ψ) ⁿ$ , where ψ is a bounded plurisubharmonic function with zero boundary values and $φ \in L^{q} ((d d^{c} ψ) ⁿ)$ , φ ≥ 0, 1 ≤ q ≤ ∞. The Dirichlet problem for the complex Monge-Ampère operator with the measure μ is studied.

Minimax nonparametric prediction

Maciej Wilczyński (2001)

Applicationes Mathematicae

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Let U₀ be a random vector taking its values in a measurable space and having an unknown distribution P and let U₁,...,Uₙ and $V ₁, . . ., V_{m}$ be independent, simple random samples from P of size n and m, respectively. Further, let $z ₁, . . ., z_{k}$ be real-valued functions defined on the same space. Assuming that only the first sample is observed, we find a minimax predictor d⁰(n,U₁,...,Uₙ) of the vector $Y^{m} = \sum_{j = 1}^{m} {(z ₁ (V_{j}), . . ., z_{k} (V_{j}))}^{T}$ with respect to a quadratic errors loss function.

Real zeros of general $L$ -functions

Alberto Perelli, Giuseppe Puglisi (1981)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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In questo lavoro vengono studiati gli zeri reali di una classe di serie di Dirichlet, che generalizzano le funzioni $L(s,\chi)$ , definite in [8], Combinando le tecniche elementari di Pintz [9] con alcuni metodi analitici si ottiene l’estensione dei classici teoremi di Hecke e Siegel.

Higher-order linear differential equations with solutions having a prescribed sequence of zeros and lying in the Dirichlet space

Li-Peng Xiao (2015)

Annales Polonici Mathematici

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The aim of this paper is to consider the following three problems:i (1) for a given uniformly q-separated sequence satisfying certain conditions, find a coefficient function A(z) analytic in the unit disc such that f”’ + A(z)f = 0 possesses a solution having zeros precisely at the points of this sequence; (2) find necessary and sufficient conditions for the differential equation $f^{(k)} + A_{k - 1} f^{(k - 1)} + \dots + A ₁ f^{'} + A ₀ f = 0$ (*) in the unit disc to be Blaschke-oscillatory; (3) find sufficient conditions on the analytic coefficients...

On the supremum of random Dirichlet polynomials

Mikhail Lifshits, Michel Weber (2007)

Studia Mathematica

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We study the supremum of some random Dirichlet polynomials $D_{N} (t) = \sum_{n = 2}^{N} ε ₙ d ₙ n^{- σ - i t}$ , where (εₙ) is a sequence of independent Rademacher random variables, the weights (dₙ) are multiplicative and 0 ≤ σ < 1/2. Particular attention is given to the polynomials $\sum_{n \in_{τ}} ε ₙ n^{- σ - i t}$ , $_{τ} = 2 \leq n \leq N : P ⁺ (n) \leq p_{τ}$ , P⁺(n) being the largest prime divisor of n. We obtain sharp upper and lower bounds for the supremum expectation that extend the optimal estimate of Halász-Queffélec, $s u p_{t \in ℝ} | \sum_{n = 2}^{N} ε ₙ n^{- σ - i t} | \approx (N^{1 - σ}) / (l o g N)$ . The proofs are entirely based on methods of stochastic processes, in particular...

IP-Dirichlet measures and IP-rigid dynamical systems: an approach via generalized Riesz products

Sophie Grivaux (2013)

Studia Mathematica

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If ${(n_{k})}_{k \geq 1}$ is a strictly increasing sequence of integers, a continuous probability measure σ on the unit circle is said to be IP-Dirichlet with respect to ${(n_{k})}_{k \geq 1}$ if $σ ̂ (\sum_{k \in F} n_{k}) \to 1$ as F runs over all non-empty finite subsets F of ℕ and the minimum of F tends to infinity. IP-Dirichlet measures and their connections with IP-rigid dynamical systems have recently been investigated by Aaronson, Hosseini and Lemańczyk. We simplify and generalize some of their results, using an approach involving generalized Riesz...

Sumsets in quadratic residues

I. D. Shkredov (2014)

Acta Arithmetica

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We describe all sets $A \subseteq_{p}$ which represent the quadratic residues $R \subseteq_{p}$ in the sense that R = A + A or R = A ⨣ A. Also, we consider the case of an approximate equality R ≈ A + A and R ≈ A ⨣ A and prove that A is then close to a perfect difference set.

Real zeros of general $L$ -functions

Alberto Perelli, Giuseppe Puglisi (1981)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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The comparison principle and Dirichlet problem in the class $_{p} (f)$ , p > 0

Pham Hoang Hiep (2006)

Annales Polonici Mathematici

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We establish the comparison principle in the class $_{p} (f)$ . The result obtained is applied to the Dirichlet problem in $_{p} (f)$ .

Another look at real quadratic fields of relative class number 1

Debopam Chakraborty, Anupam Saikia (2014)

Acta Arithmetica

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The relative class number $H_{d} (f)$ of a real quadratic field K = ℚ (√m) of discriminant d is defined to be the ratio of the class numbers of $_{f}$ and $_{K}$ , where $_{K}$ denotes the ring of integers of K and $_{f}$ is the order of conductor f given by $ℤ + f_{K}$ . R. Mollin has shown recently that almost all real quadratic fields have relative class number 1 for some conductor. In this paper we give a characterization of real quadratic fields with relative class number 1 through an elementary approach considering the...

The mean value of |L(k,χ)|² at positive rational integers k ≥ 1

Stéphane Louboutin (2001)

Colloquium Mathematicae

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Let k ≥ 1 denote any positive rational integer. We give formulae for the sums $S_{o d d} (k, f) = \sum_{χ (- 1) = - 1} | L (k, χ) | ²$ (where χ ranges over the ϕ(f)/2 odd Dirichlet characters modulo f > 2) whenever k ≥ 1 is odd, and for the sums $S_{e v e n} (k, f) = \sum_{χ (- 1) = + 1} | L (k, χ) | ²$ (where χ ranges over the ϕ(f)/2 even Dirichlet characters modulo f>2) whenever k ≥ 1 is even.

High Frequency limit of the Helmholtz Equations

Jean-David Benamou, François Castella, Thodoros Katsaounis, Benoît Perthame (1999-2000)

Séminaire Équations aux dérivées partielles

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We derive the high frequency limit of the Helmholtz equations in terms of quadratic observables. We prove that it can be written as a stationary Liouville equation with source terms. Our method is based on the Wigner Transform, which is a classical tool for evolution dispersive equations. We extend its use to the stationary case after an appropriate scaling of the Helmholtz equation. Several specific difficulties arise here; first, the identification of the source term (which does not...

Porous Medium Type Equations with a Quadratic Gradient Term

Daniela Giachetti, Giulia Maroscia (2007)

Bollettino dell'Unione Matematica Italiana

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We show an existence result for the Cauchy-Dirichlet problem in $Q_{T}=\Omega\times(0,T)$ for parabolic equations with degenerate principal part (of porous medium type) with a lower order term having a quadratic growth with respect to the gradient. The right hand side of the equation $f$ and the initial datum $u_{0}$ are bounded nonnegative functions.

Positivity of quadratic base change $L$ -functions

Hervé Jacquet, Chen Nan (2001)

Bulletin de la Société Mathématique de France

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We show that certain quadratic base change $L$ -functions for $Gl (2)$ are non-negative at their center of symmetry.

About the generating function of a left bounded integer-valued random variable

Charles Delorme, Jean-Marc Rinkel (2008)

Bulletin de la Société Mathématique de France

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We give a relation between the sign of the mean of an integer-valued, left bounded, random variable $X$ and the number of zeros of $1 - Φ (z)$ inside the unit disk, where $Φ$ is the generating function of $X$ , under some mild conditions