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Harald Helfgott

Publications and source records attributed to Harald Helfgott.

7 recordsLinked to original sources
arXiv2019

Primos, paridad y análisis

Distinguir entre enteros con un número par o impar de divisores primos es una de las tareas más difíciles en la teoría analítica de números. Un trabajo reciente de Matomäki y Radziwiłł muestra que, en promedio, ambos existen con la misma frecuencia aún en intervalos muy cortos. Este avance ya ha tenido varias aplicaciones importantes en las manos de Matomäki, Radziwiłł, Tao y Teräväinen. Explicaremos en detalle una prueba completa del resultado original de Matomäki y Radziwiłł, así como de varias aplicaciones. ----- To distinguish between integers with an even or an odd number of prime factors is one of the most difficult tasks in Analytic Number Theory. A recent work by Matomäki and Radziwiłł shows that, in average, both types of integers appear with the same frequency even in very short intervals. This breakthrough has already had several applications in the hands of Matomäki, Radziwiłł, Tao and Teräväinen. We explain in detail the complete proof of both the original result by Matomäki and Radziwiłł and of some of its applications.

math.NT↗
arXiv2018

Machine-Assisted Proofs (ICM 2018 Panel)

This submission to arXiv is the report of a panel session at the 2018 International Congress of Mathematicians (Rio de Janeiro, August). It is intended that, while v1 is that report, this stays a living document containing the panelists', and others', reflections on the topic.

math.HO↗
arXiv2012

Bounds on the diameter of Cayley graphs of the symmetric group

In this paper we are concerned with the conjecture that, for any set of generators S of the symmetric group of degree n, the word length in terms of S of every permutation is bounded above by a polynomial of n. We prove this conjecture for sets of generators containing a permutation fixing at least 37% of the points.

math.GR↗
arXiv2004

On the square-free sieve

We improve on the best available bounds for the square-free sieve and provide a general framework for its applicability. The failure of the local-to-global principle allows us to obtain results better than those reached by a classical sieve-based approach. Techniques involving sphere-packing yield upper bounds on the number of integer and rational points on curves of positive genus.

math.NT↗
arXiv2003

Root numbers and the parity problem

Let E be a one-parameter family of elliptic curves over a number field. It is natural to expect the average root number of the curves in the family to be zero. All known counterexamples to this folk conjecture occur for families obeying a certain degeneracy condition. We prove that the average root number is zero for a large class of families of elliptic curves of fairly general type. Furthermore, we show that any non-degenerate family E has average root number 0, provided that two classical arithmetical conjectures hold for two homogeneous polynomials with integral coefficients constructed explicitly in terms of E. The first such conjecture -- commonly associated with Chowla -- asserts the equidistribution of the parity of the number of primes dividing the integers represented by a polynomial. We prove the conjecture for homogeneous polynomials of degree 3. The second conjecture used states that any non-constant homogeneous polynomial yields to a square-free sieve. We sharpen the existing bounds on the known cases by a sieve refinement and a new approach combining height functions, sphere packings and sieve methods.

math.NT↗
arXiv1998

Enumeration of tilings of diamonds and hexagons with defects

We show how to count tilings of Aztec diamonds and hexagons with defects using determinants. In several cases these determinants can be evaluated in closed form. In particular, we obtain solutions to problems 1, 2, and 10 in James Propp's list of problems on enumeration of matchings.

math.CO↗