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

Liqun Hu

Displaying similar documents to “An asymptotic formula related to the divisors of the quaternary quadratic form”

The sum of divisors of a quadratic form

Lilu Zhao (2014)

Acta Arithmetica

Similarity:

We study the sum τ of divisors of the quadratic form m₁² + m₂² + m₃². Let $S ₃ (X) = \sum_{1 \leq m ₁, m ₂, m ₃ \leq X} τ (m ₁ ² + m ₂ ² + m ₃ ²)$ . We obtain the asymptotic formula S₃(X) = C₁X³logX + C₂X³ + O(X²log⁷X), where C₁,C₂ are two constants. This improves upon the error term $O_{ε} (X^{8 / 3 + ε})$ obtained by Guo and Zhai (2012).

On reduced Arakelov divisors of real quadratic fields

Ha Thanh Nguyen Tran (2016)

Acta Arithmetica

Similarity:

We generalize the concept of reduced Arakelov divisors and define C-reduced divisors for a given number C ≥ 1. These C-reduced divisors have remarkable properties, similar to the properties of reduced ones. We describe an algorithm to test whether an Arakelov divisor of a real quadratic field F is C-reduced in time polynomial in $l o g | Δ_{F} |$ with $Δ_{F}$ the discriminant of F. Moreover, we give an example of a cubic field for which our algorithm does not work.

Sumsets in quadratic residues

I. D. Shkredov (2014)

Acta Arithmetica

Similarity:

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.

Positivity of quadratic base change $L$ -functions

Hervé Jacquet, Chen Nan (2001)

Bulletin de la Société Mathématique de France

Similarity:

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

Sum of higher divisor function with prime summands

Yuchen Ding, Guang-Liang Zhou (2023)

Czechoslovak Mathematical Journal

Similarity:

Let $l ⩾ 2$ be an integer. Recently, Hu and Lü offered the asymptotic formula for the sum of the higher divisor function $\sum_{1 ⩽ n_{1}, n_{2}, ..., n_{l} ⩽ x^{1 / 2}} τ_{k} (n_{1}^{2} + n_{2}^{2} + \dots + n_{l}^{2}),$ where $τ_{k} (n)$ represents the $k$ th divisor function. We give the Goldbach-type analogy of their result. That is to say, we investigate the asymptotic behavior of the sum $\sum_{1 ⩽ p_{1}, p_{2}, ..., p_{l} ⩽ x} τ_{k} (p_{1} + p_{2} + \dots + p_{l}),$ where $p_{1}, p_{2}, \dots, p_{l}$ are prime variables.

Another look at real quadratic fields of relative class number 1

Debopam Chakraborty, Anupam Saikia (2014)

Acta Arithmetica

Similarity:

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...

A note on the distribution of angles associated to indefinite integral binary quadratic forms

Dragan Đokić (2019)

Czechoslovak Mathematical Journal

Similarity:

To each indefinite integral binary quadratic form $Q$ , we may associate the geodesic in $ℍ$ through the roots of quadratic equation $Q (x, 1)$ . In this paper we study the asymptotic distribution (as discriminant tends to infinity) of the angles between these geodesics and one fixed vertical geodesic which intersects all of them.

Asymptotic behavior of a sequence defined by iteration with applications

Stevo Stević (2002)

Colloquium Mathematicae

Similarity:

We consider the asymptotic behavior of some classes of sequences defined by a recurrent formula. The main result is the following: Let f: (0,∞)² → (0,∞) be a continuous function such that (a) 0 < f(x,y) < px + (1-p)y for some p ∈ (0,1) and for all x,y ∈ (0,α), where α > 0; (b) $f (x, y) = p x + (1 - p) y - \sum_{s = m}^{\infty} s (x, y)$ uniformly in a neighborhood of the origin, where m > 1, $_{s} (x, y) = \sum_{i = 0}^{s} a_{i, s} x^{s - i} y^{i}$ ; (c) $ₘ (1, 1) = \sum_{i = 0}^{m} a_{i, m} > 0$ . Let x₀,x₁ ∈ (0,α) and $x_{n + 1} = f (x ₙ, x_{n - 1})$ , n ∈ ℕ. Then the sequence (xₙ) satisfies the following asymptotic formula: $x ₙ \sim {((2 - p) / ((m - 1) \sum_{i = 0}^{m} a_{i, m}))}^{1 / (m - 1)} 1 / \sqrt[m - 1]{n}$ .

Asymptotic forms of solutions of perturbed half-linear ordinary differential equations

Sokea Luey, Hiroyuki Usami (2021)

Archivum Mathematicum

Similarity:

Asymptotic forms of solutions of half-linear ordinary differential equation $(| u^{'} |^{α - 1} u^{'})^{'} = α (1 + b (t)) {| u |}^{α - 1} u$ are investigated under a smallness condition and some signum conditions on $b (t)$ . When $α = 1$ , our results reduce to well-known ones for linear ordinary differential equations.

Envelope functions and asymptotic structures in Banach spaces

Bünyamin Sarı (2004)

Studia Mathematica

Similarity:

We introduce a notion of disjoint envelope functions to study asymptotic structures of Banach spaces. The main result gives a new characterization of asymptotic- $ℓ_{p}$ spaces in terms of the $ℓ_{p}$ -behavior of “disjoint-permissible” vectors of constant coefficients. Applying this result to Tirilman spaces we obtain a negative solution to a conjecture of Casazza and Shura. Further investigation of the disjoint envelopes leads to a finite-representability result in the spirit of the Maurey-Pisier...

Minimal $𝒮$ -universality criteria may vary in size

Noam D. Elkies, Daniel M. Kane, Scott Duke Kominers (2013)

Journal de Théorie des Nombres de Bordeaux

Similarity:

In this note, we give simple examples of sets $𝒮$ of quadratic forms that have minimal $𝒮$ -universality criteria of multiple cardinalities. This answers a question of Kim, Kim, and Oh [KKO05] in the negative.

Porous Medium Type Equations with a Quadratic Gradient Term

Daniela Giachetti, Giulia Maroscia (2007)

Bollettino dell'Unione Matematica Italiana

Similarity:

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.

Asymptotic integration of differential equations with singular $p$ -Laplacian

Milan Medveď, Eva Pekárková (2016)

Archivum Mathematicum

Similarity:

In this paper we deal with the problem of asymptotic integration of nonlinear differential equations with $p -$ Laplacian, where $1 < p < 2$ . We prove sufficient conditions under which all solutions of an equation from this class are converging to a linear function as $t \to \infty$ .