On reduced Arakelov divisors of real quadratic fields

Ha Thanh Nguyen Tran

Displaying similar documents to “On reduced Arakelov divisors of real quadratic fields”

The sum of divisors of a quadratic form

Lilu Zhao (2014)

Acta Arithmetica

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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).

An improvement of Euclid's algorithm

Zítko, Jan, Kuřátko, Jan

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The paper introduces the calculation of a greatest common divisor of two univariate polynomials. Euclid’s algorithm can be easily simulated by the reduction of the Sylvester matrix to an upper triangular form. This is performed by using $c$ - $s$ transformation and $Q R$ -factorization methods. Both procedures are described and numerically compared. Computations are performed in the floating point environment.

Stein open subsets with analytic complements in compact complex spaces

Jing Zhang (2015)

Annales Polonici Mathematici

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Let Y be an open subset of a reduced compact complex space X such that X - Y is the support of an effective divisor D. If X is a surface and D is an effective Weil divisor, we give sufficient conditions so that Y is Stein. If X is of pure dimension d ≥ 1 and X - Y is the support of an effective Cartier divisor D, we show that Y is Stein if Y contains no compact curves, $H^{i} (Y,_{Y}) = 0$ for all i > 0, and for every point x₀ ∈ X-Y there is an n ∈ ℕ such that $Φ_{| n D |}^{- 1} (Φ_{| n D |} (x ₀)) \cap Y$ is empty or has dimension 0, where $Φ_{| n D |}$ ...

Calculation of the greatest common divisor of perturbed polynomials

Zítko, Jan, Eliaš, Ján

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The coefficients of the greatest common divisor of two polynomials $f$ and $g$ (GCD $(f, g)$ ) can be obtained from the Sylvester subresultant matrix $S_{j} (f, g)$ transformed to lower triangular form, where $1 \leq j \leq d$ and $d =$ deg(GCD $(f, g)$ ) needs to be computed. Firstly, it is supposed that the coefficients of polynomials are given exactly. Transformations of $S_{j} (f, g)$ for an arbitrary allowable $j$ are in details described and an algorithm for the calculation of the GCD $(f, g)$ is formulated. If inexact polynomials are given, then an approximate...

The mean square of the divisor function

Chaohua Jia, Ayyadurai Sankaranarayanan (2014)

Acta Arithmetica

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Let d(n) be the divisor function. In 1916, S. Ramanujan stated without proof that $\sum_{n \leq x} d ² (n) = x P (l o g x) + E (x)$ , where P(y) is a cubic polynomial in y and $E (x) = O (x^{3 / 5 + ε})$ , with ε being a sufficiently small positive constant. He also stated that, assuming the Riemann Hypothesis (RH), $E (x) = O (x^{1 / 2 + ε})$ . In 1922, B. M. Wilson proved the above result unconditionally. The direct application of the RH would produce $E (x) = O (x^{1 / 2} (l o g x) ⁵ l o g l o g x)$ . In 2003, K. Ramachandra and A. Sankaranarayanan proved the above result without any assumption. In this paper, we prove $E (x) = O (x^{1 / 2} (l o g x) ⁵)$ . ...

On the existence of dimension zero divisors in algebraic function fields defined over $_{q}$

S. Ballet, C. Ritzenthaler, R. Rolland (2010)

Acta Arithmetica

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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...

Factorization of matrices associated with classes of arithmetical functions

Shaofang Hong (2003)

Colloquium Mathematicae

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Let f be an arithmetical function. A set S = x₁,..., xₙ of n distinct positive integers is called multiple closed if y ∈ S whenever x|y|lcm(S) for any x ∈ S, where lcm(S) is the least common multiple of all elements in S. We show that for any multiple closed set S and for any divisor chain S (i.e. x₁|...|xₙ), if f is a completely multiplicative function such that (f*μ)(d) is a nonzero integer whenever d|lcm(S), then the matrix $(f (x_{i}, x_{i}))$ having f evaluated at the greatest common divisor $(x_{i}, x_{i})$ of...