Displaying similar documents to “On the error term of the logarithm of the lcm of a quadratic sequence”

Denominators of Igusa class polynomials

Kristin Lauter, Bianca Viray (2014)

Publications mathématiques de Besançon

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In [], the authors proved an explicit formula for the arithmetic intersection number CM ( K ) . G 1 on the Siegel moduli space of abelian surfaces, under some assumptions on the quartic CM field K . These intersection numbers allow one to compute the denominators of Igusa class polynomials, which has important applications to the construction of genus 2 curves for use in cryptography. One of the main tools in the proof was a previous result of the authors [] generalizing the singular moduli formula...

Invariant measures and long-time behavior for the Benjamin-Ono equation

Yu Deng, Nikolay Tzvetkov, Nicola Visciglia (2014)

Journées Équations aux dérivées partielles

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We summarize the main ideas in a series of papers ([], [], [], []) devoted to the construction of invariant measures and to the long-time behavior of solutions of the periodic Benjamin-Ono equation.

On Gelfond’s conjecture about the sum of digits of prime numbers

Joël Rivat (2009)

Journal de Théorie des Nombres de Bordeaux

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The goal of this paper is to outline the proof of a conjecture of Gelfond [] (1968) in a recent work in collaboration with Christian Mauduit [] concerning the sum of digits of prime numbers, reflecting the lecture given in Edinburgh at the Journées Arithmétiques 2007.

On prime values of reducible quadratic polynomials

W. Narkiewicz, T. Pezda (2002)

Colloquium Mathematicae

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It is shown that Dickson’s Conjecture about primes in linear polynomials implies that if f is a reducible quadratic polynomial with integral coefficients and non-zero discriminant then for every r there exists an integer N r such that the polynomial f ( X ) / N r represents at least r distinct primes.

Lefschetz Fibrations and real Lefschetz fibrations

Nermin Salepci (2014)

Winter Braids Lecture Notes

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This note is based on the lectures that I have given during the winter school Winter Braids IV, School on algebraic and topological aspects of braid groups held in Dijon on 10 - 13 February 2014. The aim of series of three lectures was to give an overview of geometrical and topological properties of 4-dimensional Lefschetz fibrations. Meanwhile, I could briefly introduce real Lefschetz fibrations, fibrations which have certain symmetry, and could present some...

The geometry of dimer models

David Cimasoni (2014)

Winter Braids Lecture Notes

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This is an expanded version of a three-hour minicourse given at the winterschool held in Dijon in February 2014. The aim of these lectures was to present some aspects of the dimer model to a geometrically minded audience. We spoke neither of braids nor of knots, but tried to show how several geometric tools that we know and love (e.g. (co)homology, spin structures, real algebraic curves) can be applied to very natural problems in combinatorics and statistical physics. These lecture...

Gibbs-Markov-Young structures, ,

Carla L. Dias (2012)

ESAIM: Proceedings

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We discuss the geometric structures defined by Young in [9, 10], which are used to prove the existence of an ergodic absolutely continuous invariant probability measure and to study the decay of correlations in expanding or hyperbolic systems on large parts.

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...

Sumsets in quadratic residues

I. D. Shkredov (2014)

Acta Arithmetica

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We describe all sets A ⊆ p which represent the quadratic residues R ⊆ 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.

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 ) = ∑ 1 ≤ m ₁ , m ₂ , m ₃ ≤ 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).

Quadratic modular symbols on Shimura curves

Pilar Bayer, Iván Blanco-Chacón (2013)

Journal de Théorie des Nombres de Bordeaux

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We introduce the concept of modular symbol and study how these symbols are related to p -adic L -functions. These objects were introduced in [] in the case of modular curves. In this paper, we discuss a method to attach quadratic modular symbols and quadratic p -adic L -functions to more general Shimura curves.

High-order phase transitions in the quadratic family

Daniel Coronel, Juan Rivera-Letelier (2015)

Journal of the European Mathematical Society

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We give the first example of a transitive quadratic map whose real and complex geometric pressure functions have a high-order phase transition. In fact, we show that this phase transition resembles a Kosterlitz-Thouless singularity: Near the critical parameter the geometric pressure function behaves as x ↦ exp ( – x – 2 ) near x = 0 , before becoming linear. This quadratic map has a non-recurrent critical point, so it is non-uniformly hyperbolic in a strong sense.

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...