ID: 1502.05062

Distribution of squares modulo a composite number

February 17, 2015

View on ArXiv

Similar papers 2

On the distribution of modular square roots of primes

September 8, 2020

86% Match
Ilya D. Shkredov, Igor E. Shparlinski, Alexandru Zaharescu
Number Theory

We use recent bounds on bilinear sums with modular square roots to study the distribution of solutions to congruences $x^2 \equiv p \pmod q$ with primes $p\le P$ and integers $q \le Q$. This can be considered as a combined scenario of Duke, Friedlander and Iwaniec with averaging only over the modulus $q$ and of Dunn, Kerr, Shparlinski and Zaharescu with averaging only over $p$.

Find SimilarView on arXiv

Counting coprime pairs in random squares

March 19, 2024

86% Match
José L. Fernández, Pablo Fernández
Number Theory
Probability

Extending the classical Dirichlet's density theorem on coprime pairs, in this paper we describe completely the probability distribution of the number of coprime pairs in random squares of fixed side length in the lattice $\mathbb{N}^2$. The limit behaviour of this distribution as the side length of the random square tends to infinity is also considered.

Find SimilarView on arXiv

On square-free values of large polynomials over the rational function field

May 25, 2016

85% Match
Dan Carmon, Alexei Entin
Number Theory

We investigate the density of square-free values of polynomials with large coefficients over the rational function field $\mathbb{F}_q[t]$. Some interesting questions answered as special cases of our results include the density of square-free polynomials in short intervals, and an asymptotic for the number of representations of a large polynomial $N$ as a sum of a small $k$-th power and a square-free polynomial.

Find SimilarView on arXiv

Additive Correlation and the Inverse Problem for the Large Sieve

June 21, 2017

85% Match
Brandon Hanson
Number Theory

Let $A\subset [1,N]$ be a set of positive integers with $|A|\gg \sqrt N$. We show that if avoids about $p/2$ residue classes modulo $p$ for each prime $p$, the $A$ must correlate additively with the squares $S=\{n^2:1\leq n\leq \sqrt N\}$, in the sense that we have the additive energy estimate $E(A,S)\gg N\log N$. This is, in a sense, optimal.

Find SimilarView on arXiv

On the square-free sieve

September 5, 2003

85% Match
Harald Helfgott
Number Theory
Algebraic Geometry

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.

Find SimilarView on arXiv

Large Sieve Inequalities for Characters to Square Moduli

August 7, 2005

85% Match
Liangyi Zhao
Number Theory

In this paper, we develop a large sieve type inequality with characters to square moduli. One expects that the result should be weaker than the classical inequality, but, conjecturally at least, not by much. The method is generalizable to higher power moduli.

Find SimilarView on arXiv

The distribution of spacings between quadratic residues

December 8, 1998

85% Match
P. Kurlberg, Z. Rudnick
Number Theory
Mathematical Physics

We study the distribution of spacings between squares modulo q, where q is square-free and highly composite, in the limit as the number of prime factors of q goes to infinity. We show that all correlation functions are Poissonian, which among other things, implies that the spacings between nearest neighbors, normalized to have unit mean, have an exponential distribution.

Find SimilarView on arXiv

On sums of squares of primes II

February 24, 2009

84% Match
Glyn Harman, Angel Kumchev
Number Theory

In this paper we continue our study, begun in part I, of the exceptional set of integers, not restricted by elementary congruence conditions, which cannot be represented as sums of three or four squares of primes. We correct a serious oversight in our first paper, but make further progress on the exponential sums estimates needed, together with an embellishment of the previous sieve technique employed. This leads to an improvement in our bounds for the maximal size of the exc...

Find SimilarView on arXiv

Sums of almost equal squares of primes

September 26, 2011

84% 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

The large sieve with square norm moduli in Z[i]

November 8, 2015

84% Match
Stephan Baier
Number Theory

We prove a large sieve inequality for square norm moduli in Z[i].

Find SimilarView on arXiv