ID: 1601.04754

Prescribing the binary digits of squarefree numbers and quadratic residues

January 18, 2016

View on ArXiv

Similar papers 3

Hilbert cubes in arithmetic sets

October 18, 2015

82% Match
Rainer Dietmann, Christian Elsholtz
Number Theory
Combinatorics

We show upper bounds on the maximal dimension $d$ of Hilbert cubes $H=a_0+\{0,a_1\}+\cdots + \{0, a_d\}\subset S \cap [1, N]$ in several sets $S$ of arithmetic interest such as the squares, powerful numbers and pure powers.

Find SimilarView on arXiv

Additive problems with almost prime squares

November 2, 2021

82% Match
Valentin Blomer, Lasse Grimmelt, ... , Myerson Simon L. Rydin
Number Theory

We show that every sufficiently large integer is a sum of a prime and two almost prime squares, and also a sum of a smooth number and two almost prime squares. The number of such representations is of the expected order of magnitude. We likewise treat representations of shifted primes p-1 as sums of two almost prime squares. The methods involve a combination of analytic, automorphic and algebraic arguments to handle representations by restricted binary quadratic forms with a ...

Find SimilarView on arXiv

Sumsets avoiding squarefree integers

May 6, 2011

82% Match
Jan-Christoph Schlage-Puchta
Number Theory

We describe the structure of a set of integers $A$ of positive density $\delta$, such that $A+A$ contains no squarefree integer. It turns out that the behaviour changes abruptly at the values $\delta_0=1/4-\frac{2}{\pi^2}=0.0473...$ and $\delta_1=1/18$.

Find SimilarView on arXiv

Short intervals asymptotic formulae for binary problems with primes and powers, II: density $1$

April 18, 2015

82% Match
Alessandro Languasco, Alessandro Zaccagnini
Number Theory

We prove that suitable asymptotic formulae in short intervals hold for the problems of representing an integer as a sum of a prime square and a square, or a prime square. Such results are obtained both assuming the Riemann Hypothesis and in the unconditional case.

Find SimilarView on arXiv

The frequency and the structure of large character sums

October 29, 2014

82% Match
Jonathan Bober, Leo Goldmakher, ... , Koukoulopoulos Dimitris
Number Theory

Let $M(\chi)$ denote the maximum of $|\sum_{n\le N}\chi(n)|$ for a given non-principal Dirichlet character $\chi \pmod q$, and let $N_\chi$ denote a point at which the maximum is attained. In this article we study the distribution of $M(\chi)/\sqrt{q}$ as one varies over characters $\pmod q$, where $q$ is prime, and investigate the location of $N_\chi$. We show that the distribution of $M(\chi)/\sqrt{q}$ converges weakly to a universal distribution $\Phi$, uniformly throughou...

Find SimilarView on arXiv

Sums of almost equal squares of primes

September 26, 2011

82% Match
Angel Kumchev, Taiyu Li
Number Theory

We study the representations of large integers $n$ as sums $p_1^2 + ... + p_s^2$, where $p_1,..., p_s$ are primes with $| p_i - (n/s)^{1/2} | \le n^{\theta/2}$, for some fixed $\theta < 1$. When $s = 5$ we use a sieve method to show that all sufficiently large integers $n \equiv 5 \pmod {24}$ can be represented in the above form for $\theta > 8/9$. This improves on earlier work by Liu, L\"{u} and Zhan, who established a similar result for $\theta > 9/10$. We also obtain estim...

Find SimilarView on arXiv

On the density or measure of sets and their sumsets in the integers or the circle

May 20, 2019

82% Match
Pierre-Yves Bienvenu, François Hennecart
Number Theory

Let $\mathrm{d}(A)$ be the asymptotic density (if it exists) of a sequence of integers $A$. For any real numbers $0\leq\alpha\leq\beta\leq 1$, we solve the question of the existence of a sequence $A$ of positive integers such that $\mathrm{d}(A)=\alpha$ and $\mathrm{d}(A+A)=\beta$. More generally we study the set of $k$-tuples $(\mathrm{d}(iA))_{1\leq i\leq k}$ for $A\subset \mathbb{N}$. This leads us to introduce subsets defined by diophantine constraints inside a random set...

Find SimilarView on arXiv

Sums of four squareful numbers

April 14, 2021

82% Match
Alec Shute
Number Theory

We find an asymptotic formula for the number of primitive vectors $(z_1,\ldots,z_4)\in (\mathbb{Z}_{\neq 0})^4$ such that $z_1,\ldots, z_4$ are all squareful and bounded by $B$, and $z_1+\cdots + z_4 = 0$. Our result agrees in the power of $B$ and $\log B$ with the Campana-Manin conjecture of Pieropan, Smeets, Tanimoto and V\'{a}rilly-Alvarado.

Find SimilarView on arXiv

The distribution of large quadratic character sums and applications

December 6, 2022

82% Match
Youness Lamzouri
Number Theory

In this paper, we investigate the distribution of the maximum of character sums over the family of primitive quadratic characters attached to fundamental discriminants $|d|\leq x$. In particular, our work improves results of Montgomery and Vaughan, and gives strong evidence that the Omega result of Bateman and Chowla for quadratic character sums is optimal. We also obtain similar results for real characters with prime discriminants up to $x$, and deduce the interesting conseq...

Find SimilarView on arXiv

An invitation to additive prime number theory

December 10, 2004

82% Match
A. V. University of Texas Kumchev, D. I. Plovdiv University Tolev
Number Theory

The main purpose of this survey is to introduce an inexperienced reader to additive prime number theory and some related branches of analytic number theory. We state the main problems in the field, sketch their history and the basic machinery used to study them, and try to give a representative sample of the directions of current research.

Find SimilarView on arXiv