50 citations · 353 across the 37 of their papers we have counts for
23 papers · 1 filter
Optimal bounds for sums of non-negative arithmetic functions
Andrés Chirre, Harald Andrés Helfgott
Let be a Dirichlet series admitting meromorphic continuation to the complex plane. Assume we know the location of the poles of with $|\Im s| \leq…
Optimal bounds for sums of bounded arithmetic functions
Andrés Chirre, Harald Andrés Helfgott
Let be a Dirichlet series with meromorphic continuation. Say we are given information on the poles of with for some large constan…
Expansion, divisibility and parity: an explanation
Harald Andrés Helfgott
After seeing how questions on the finer distribution of prime factorization -- considered inaccessible until recently -- reduce to bounding the norm of an operator defined on a gra…
Expansion, divisibility and parity
Harald Andrés Helfgott, Maksym Radziwiłł
Let be a set of primes, where . Let . Let be such that $\log H \leq (\…
Summing : a faster elementary algorithm
Harald A. Helfgott, Lola Thompson
We present a new elementary algorithm that takes \[ \mathrm{time} \ \ O_ε\left(x^{\frac{3}{5}} (\log x)^{\frac{3}{5}+ε} \right) \ \ \mathrm{and}\ \ \mathrm{space} \ \ O\left(x^{\fr…
Optimality for the two-parameter quadratic sieve
Emanuel Carneiro, Andrés Chirre, Harald Andrés Helfgott +1
We study the two-parameter quadratic sieve for a general test function. We prove, under some very general assumptions, that the function considered by Barban and Vehov [BV68] and G…
An improved sieve of Eratosthenes
Harald Andres Helfgott
We show how to carry out a sieve of Eratosthenes up to N in space O(N^{1/3} (log N)^{2/3}) and time O(N log N). These bounds constitute an improvement over the usual versions of th…
The ternary Goldbach problem
Harald Andres Helfgott
The ternary Goldbach conjecture (or three-prime conjecture) states that every odd number greater than 5 can be written as the sum of three primes. The purpose of this book is to gi…
The ternary Goldbach problem
Harald Andrés Helfgott
The ternary Goldbach conjecture, or three-primes problem, states that every odd number greater than can be written as the sum of three primes. The conjecture, posed in 1742…
Numerical Verification of the Ternary Goldbach Conjecture up to 8.875e30
H. A. Helfgott, David J. Platt
We describe a computation that confirms the ternary Goldbach Conjecture up to 8,875,694,145,621,773,516,800,000,000,000 (>8.875e30).
Major arcs for Goldbach's problem
H. A. Helfgott
The ternary Goldbach conjecture states that every odd number is the sum of three primes. The estimation of the Fourier series and related sums has…
Minor arcs for Goldbach's problem
H. A. Helfgott
The ternary Goldbach conjecture states that every odd number n>=7 is the sum of three primes. The estimation of sums of the form \sum_{p\leq x} e(αp), α= a/q + O(1/q^2), has been a…
Square-free values of f(p), f cubic
H. A. Helfgott
Let f\in Z[x], deg(f)=3. Assume that f does not have repeated roots. Assume as well that, for every prime q, the inequality f(x)\not\equiv 0 mod q^2 has at least one solution in (Z…
Improving Roth's theorem in the primes
Harald Andres Helfgott, Anne de Roton
Let A be a subset of the primes. Let δ_P(N) = \frac{|\{n\in A: n\leq N\}|}{|\{\text{ prime}: n\leq N\}|}. We prove that, if δ_P(N)\geq C \frac{\log \log \log N}{(\log \log N)^{1…
Power-free values, repulsion between points, differing beliefs and the existence of error
Harald Andres Helfgott
Let f be a cubic polynomial. Then there are infinitely many primes p such that f(p) is square-free.
The parity problem for irreducible cubic forms
H. A. Helfgott
Let f\in \mathbb{Z}[x,y] be an irreducible homogeneous polynomial of degree 3. We show that f(x,y) has an even number of prime factors as often as an odd number of prime factors.
Power-free values, large deviations, and integer points on irrational curves
H. A. Helfgott
Let be a polynomial of degree without roots of multiplicity or . Erdős conjectured that, if satisfies the necessary local…
Root numbers and ranks in positive characteristic
B. Conrad, K. Conrad, H. Helfgott
For a global field K and an elliptic curve E_eta over K(T), Silverman's specialization theorem implies that rank(E_eta(K(T))) <= rank(E_t(K)) for all but finitely many t in P^1(K).…
The parity problem for reducible cubic forms
H. A. Helfgott
Let f in Z[x,y] be a reducible homoegeneous polynomial of degree 3. We show that f(x,y) has an even number of prime factors as often as an odd number of prime factors.
On the behaviour of root numbers in families of elliptic curves
H. A. Helfgott
Let E be a one-parameter family of elliptic curves over Q. We prove that the average root number is zero for a large class of families of elliptic curves of fairly general type. Fu…
Integral points on elliptic curves and 3-torsion in class groups
H. A. Helfgott, A. Venkatesh
We give new bounds for the number of integral points on elliptic curves. The method may be said to interpolate between approaches via diophantine techniques ([BP], [HBR]) and metho…
On the square-free sieve
Harald Helfgott
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 o…
Root numbers and the parity problem
Harald Helfgott
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 counterexa…