The distribution of smooth numbers in arithmetic progressions

Transactions of the American Mathematical Society, 2021

We establish new Bombieri-Vinogradov type estimates for a wide class of multiplicative arithmetic functions and derive several applications, including: a new proof of a recent estimate by Drappeau and Topacogullari for arithmetical correlations; a theorem of Erdős-Wintner type with support equal to the level set of an additive function at shifted argument; and a law of iterated logarithm for the distribution of prime factors of integers weighted by τ (n − 1) where τ denotes the divisor function.

arXiv: Number Theory, 2020

We establish new Bombieri-Vinogradov type estimates for a wide class of multiplicative arithmetic functions and derive several applications, including: a new proof of a recent estimate by Drappeau and Topacogullari for arithmetical correlations; a theorem of Erd{\H o}s-Wintner type with support equal to the level set of an additive function at shifted argument; and a law of iterated logarithm for the distribution of prime factors of integers weighted by $\tau(n-1)$ where $\tau$ denotes the divisor function.

2004

We describe some of the machinery behind recent progress in establishing infinitely many arithmetic progressions of length k in various sets of integers, in particular in arbitrary dense subsets of the integers, and in the primes.

Transactions of the American Mathematical Society, 1986

The number Ψ ( x , y ) \Psi (x,y) of integers ≤ x \leq x and free of prime factors > y > y has been given satisfactory estimates in the regions y ≤ ( log ⁡ x ) 3 / 4 − ε y \leq {(\log x)^{3/4 - \varepsilon }} and y > exp ⁡ { ( log ⁡ log ⁡ x ) 5 / 3 + ε } y > \exp \{ {(\log \log x)^{5/3 + \varepsilon }}\} . In the intermediate range, only very crude estimates have been obtained so far. We close this "gap" and give an expression which approximates Ψ ( x , y ) \Psi (x,y) uniformly for x ≥ y ≥ 2 x \geq y \geq 2 within a factor 1 + O ( ( log ⁡ y ) / ( log ⁡ x ) + ( log ⁡ y ) / y ) 1 + O((\log y)/(\log x) + (\log y)/y) . As an application, we derive a simple formula for Ψ ( c x , y ) / Ψ ( x , y ) \Psi (cx,y)/\Psi (x,y) , where 1 ≤ c ≤ y 1 \leq c \leq y . We also prove a short interval estimate for Ψ ( x , y ) \Psi (x,y) .

Journal of the London Mathematical Society, 1990

We nd in nitely many pairs of coprime integers, a and q, such that the least prime a (mod q) is unusually large. In so doing we also consider the question of approximating rationals by other rationals with smaller and coprime denominators.

2006

We describe some of the machinery behind recent progress in establishing infinitely many arithmetic progressions of length k in various sets of integers, in particular in arbitrary dense subsets of the integers, and in the primes.

Mathematics of Computation, 1996

Strengthening work of Rosser, Schoenfeld, and McCurley, we establish explicit Chebyshev-type estimates in the prime number theorem for arithmetic progressions, for all moduli k ≤ 72 k \le 72 and other small moduli.

Discrete Mathematics & Theoretical Computer Science, 2006

We solve a problem by V. I. Arnold dealing with "how random" modular arithmetic progressions can be. After making precise how Arnold proposes to measure the randomness of a modular sequence, we show that this measure of randomness takes a simplified form in the case of arithmetic progressions. This simplified expression is then estimated using the methodology of dynamical analysis, which operates with tools coming from dynamical systems theory. In conclusion, this study shows that modular arithmetic progressions are far from behaving like purely random sequences, according to Arnold's definition.

Journal of Combinatorial Theory, Series A, 1982

The maximal density attainable by a sequence S of positive integers having the property that the sum of any two elements of S is never a square is studied. J. P. Massias exhibited such a sequence with density g; it consists of 11 residue classes (mod 32) such that the sum of any two such residue classes is not congruent to a square (mod 32). It is shown that for any positive integer n, one cannot find more than gn residue classes (mod n) such that the sum of any two is never congruent to a square (mod n). Thus Massias' example has maximal density among those sequences S made up of a finite set of (infinite) arithmetic progressions. A companion paper will bound the maximal density of an arbitrary such sequence S.

Indagationes Mathematicae, 2006