ID: math/9812047

The distribution of spacings between quadratic residues, II

December 8, 1998

View on ArXiv

Similar papers 2

The distribution of spacings between the fractional parts of $\boldsymbol{n^d\alpha}$

April 10, 2020

86% Match
Martino Fassina, Sun Kim, Alexandru Zaharescu
Number Theory

We study the distribution of spacings between the fractional parts of $n^d\alpha$. For $\alpha$ of high enough Diophantine type we prove a necessary and sufficient condition for $n^d\alpha\mod 1, 1\leq n\leq N,$ to be Poissonian as $N\to \infty$ along a suitable subsequence.

Find SimilarView on arXiv

Mean Values for a Class of Arithmetic Functions in Short Intervals

July 24, 2018

85% Match
Jie UPVM Wu, Qiang UPVM Wu
Number Theory

In this paper, we shall establish a rather general asymptotic formula in short intervals for a classe of arithmetic functions and announce two applications about the distribution of divisors of square-full numbers and integers representable as sums of two squares.

Find SimilarView on arXiv

Counting coprime pairs in random squares

March 19, 2024

85% 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 numbers generated from given sets of primes II

January 6, 2023

85% Match
Gábor Román
Number Theory

We progress with the investigation started in article \cite{Roman2022}, namely the analysis of the asymptotic behaviour of $Q_{\mathcal{P}}(x)$ for different sets $\mathcal{P}$, where $Q_{\mathcal{P}}(x)$ is the element count of the set containing those positive square-free integers, which are smaller than-, or equal to $x$, and which are only divisible by the elements of $\mathcal{P}$. We study how $Q_{\mathcal{P}}(x)$ behaves when we require that $\chi(p) = 1$ must hold for...

Find SimilarView on arXiv

Distribution of Values of Real Quadratic Zeta Functions

October 29, 2000

85% Match
Joshua Holden
Number Theory
Numerical Analysis

The author has previously extended the theory of regular and irregular primes to the setting of arbitrary totally real number fields. It has been conjectured that the Bernoulli numbers, or alternatively the values of the Riemann zeta function at odd negative integers, are evenly distributed modulo p for every p. This is the basis of a well-known heuristic, given by Siegel, for estimating the frequency of irregular primes. So far, analyses have shown that if Q(\sqrt{D}) is a r...

Find SimilarView on arXiv

On the distribution of the divisor function and Hecke eigenvalues

April 6, 2014

85% Match
Stephen Lester, Nadav Yesha
Number Theory

We investigate the behavior of the divisor function in both short intervals and in arithmetic progressions. The latter problem was recently studied by \'E. Fouvry, S. Ganguly, E. Kowalski, and Ph. Michel. We prove a complementary result to their main theorem. We also show that in short intervals of certain lengths the divisor function has a Gaussian limiting distribution. The analogous problems for Hecke eigenvalues are also considered.

Find SimilarView on arXiv

Short interval results for certain prime-independent multiplicative functions

September 9, 2016

85% Match
Olivier Bordellès
Number Theory

Using recent results from the theory of integer points close to smooth curves, we give an asymptotic formula for the distribution of values of a class of integer-valued prime-independent multiplicative functions.

Find SimilarView on arXiv

Squarefree density of polynomials

September 18, 2023

85% Match
J. M. Kowalski, R. C Vaughan
Number Theory

This paper is concerned with squarefree values of polynomials and their density in large boxes centered at the origin.

Find SimilarView on arXiv

Multiplicative functions in short intervals II

July 8, 2020

85% Match
Kaisa Matomäki, Maksym Radziwiłł
Number Theory

We determine the behavior of multiplicative functions vanishing at a positive proportion of prime numbers in almost all short intervals. Furthermore we quantify "almost all" with uniform power-saving upper bounds, that is, we save a power of the suitably normalized length of the interval regardless of how long or short the interval is. Such power-saving bounds are new even in the special case of the M\"obius function. These general results are motivated by several application...

Find SimilarView on arXiv

Primes in Quadratic Progressions on Average

May 20, 2006

85% Match
Stephan Baier, Liangyi Zhao
Number Theory

In this paper, we establish a theorem on the distribution of primes in quadratic progressions on average.

Find SimilarView on arXiv