On a Waring-Goldbach problem involving squares and cubes

2019 ◽  
Vol 69 (6) ◽  
pp. 1249-1262
Author(s):  
Yuhui Liu
Keyword(s):  
Asymptotic Formula ◽  
Natural Number ◽  
Goldbach Problem ◽  
Positive Integers

AbstractLet R(n) denote the number of representations of a natural number n as the sum of two squares and four cubes of primes. In this paper, it is proved that the anticipated asymptotic formula for R(n) fails for at most $\begin{array}{}O(N^{\frac{1}{4} + \varepsilon})\end{array}$ positive integers not exceeding N.

THE EXCEPTIONAL SET IN THE POLYNOMIAL GOLDBACH PROBLEM

2011 ◽  
Vol 07 (03) ◽  
pp. 579-591 ◽  
Author(s):  
PAUL POLLACK
Keyword(s):  
Asymptotic Formula ◽  
Finite Field ◽  
Natural Number ◽  
Riemann Hypothesis ◽  
Natural Numbers ◽  
Exceptional Set ◽  
Goldbach Problem

For each natural number N, let R(N) denote the number of representations of N as a sum of two primes. Hardy and Littlewood proposed a plausible asymptotic formula for R(2N) and showed, under the assumption of the Riemann Hypothesis for Dirichlet L-functions, that the formula holds "on average" in a certain sense. From this they deduced (under ERH) that all but Oϵ(x1/2+ϵ) of the even natural numbers in [1, x] can be written as a sum of two primes. We generalize their results to the setting of polynomials over a finite field. Owing to Weil's Riemann Hypothesis, our results are unconditional.

scholarly journals ON THE ASYMPTOTIC FORMULA IN WARING'S PROBLEM: ONE SQUARE AND THREE FIFTH POWERS

2014 ◽  
Vol 57 (3) ◽  
pp. 681-692
Author(s):  
JÖRG BRÜDERN
Keyword(s):  
Asymptotic Formula ◽  
Natural Number ◽  
Asymptotic Relation ◽  
Circle Method ◽  
Precise Definition ◽  
Singular Series ◽  
Mean Square ◽  
Formula Group ◽  
Image Position ◽  
Positive Integers

1. Let r(n) denote the number of representations of the natural number n as the sum of one square and three fifth powers of positive integers. A formal use of the circle method predicts the asymptotic relation (1)$ \begin{equation*} r(n) = \frac{\Gamma(\frac32)\Gamma(\frac65)^3}{\Gamma(\frac{11}{10})} {\mathfrak s}(n) {n}^\frac1{10} (1 + o(1)) \qquad (n\to\infty). \end{equation*} $ Here ${\mathfrak s}$(n) is the singular series associated with sums of a square and three fifth powers, see (13) below for a precise definition. The main purpose of this note is to confirm (1) in mean square.

XLIX.—Generalizations of a Problem of Pillai

1949 ◽  
Vol 62 (4) ◽  
pp. 460-469
Author(s):  
L. Mirsky
Keyword(s):  
Asymptotic Formula ◽  
Error Term ◽  
Main Concern ◽  
Positive Integers ◽  
Infinite Set

I. Throughout this paper k1, …, k3 will denote s ≥ I fixed distinct positive integers. Some years ago Pillai (1936) found an asymptotic formula, with error term O(x/log x), for the number of positive integers n ≤ x such that n + k1, …, n + k3 are all square-free. I recently considered (Mirsky, 1947) the corresponding problem for r-free integers (i.e. integers not divisible by the rth power of any prime), and was able, in particular, to reduce the error term in Pillai's formula.Our present object is to discuss various generalizations and extensions of Pillai's problem. In all investigations below we shall be concerned with a set A of integers. This is any given, finite or infinite, set of integers greater than 1 and subject to certain additional restrictions which will be stated later. The elements of A will be called a-numbers, and the letter a will be reserved for them. A number which is not divisible by any a-number will be called A-free, and our main concern will be with the study of A-free numbers. Their additive properties have recently been investigated elsewhere (Mirsky, 1948), and some estimates obtained in that investigation will be quoted in the present paper.

Asymptotic formula for sum of moment mean deviation for order statistics from uniform distribution

2019 ◽  
Vol 11 (02) ◽  
pp. 1950015
Author(s):  
Rafał Kapelko
Keyword(s):  
Positive Integer ◽  
Asymptotic Formula ◽  
Natural Number ◽  
Uniform Distribution ◽  
Gamma Function ◽  
Order Statistic ◽  
Unit Interval ◽  
Mobile Sensors ◽  
Mean Deviation ◽  
Current Location

Assume that [Formula: see text] mobile sensors are thrown uniformly and independently at random with the uniform distribution on the unit interval. We study the expected sum over all sensors [Formula: see text] from [Formula: see text] to [Formula: see text] where the contribution of the [Formula: see text] sensor is its displacement from the current location to the anchor equidistant point [Formula: see text] raised to the [Formula: see text] power, when [Formula: see text] is an odd natural number. As a consequence, we derive the following asymptotic identity. Fix [Formula: see text] positive integer. Let [Formula: see text] denote the [Formula: see text] order statistic from a random sample of size [Formula: see text] from the Uniform[Formula: see text] population. Then [Formula: see text] where [Formula: see text] is the Gamma function.

scholarly journals Sums of smooth squares

2009 ◽  
Vol 145 (6) ◽  
pp. 1401-1441 ◽  
Author(s):  
V. Blomer ◽  
J. Brüdern ◽  
R. Dietmann
Keyword(s):  
Lower Bound ◽  
Asymptotic Formula ◽  
Natural Number ◽  
Related Problem ◽  
Order Of Magnitude ◽  
Sums Of Three Squares

AbstractLet R(n,θ) denote the number of representations of the natural number n as the sum of four squares, each composed only with primes not exceeding nθ/2. When θ>e−1/3 a lower bound for R(n,θ) of the expected order of magnitude is established, and when θ>365/592, it is shown that R(n,θ)>0 holds for large n. A similar result is obtained for sums of three squares. An asymptotic formula is obtained for the related problem of representing an integer as the sum of two squares and two squares composed of small primes, as above, for any fixed θ>0. This last result is the key to bound R(n,θ) from below.

scholarly journals A Note on the Number of $(k,l)$-Sum-Free Sets

10.37236/1508 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
Tomasz Schoen
Keyword(s):  
Natural Number ◽  
Positive Integers ◽  
Free Sets

A set $A\subseteq {\bf N}$ is $(k,\ell)$-sum-free, for $k,\ell\in {\bf N}$, $k>\ell$, if it contains no solutions to the equation $x_1+\dots+x_k=y_1+\dots+y_{\ell}$. Let $\rho=\rho (k-\ell)$ be the smallest natural number not dividing $k-\ell$, and let $r=r_n$, $0\le r < \rho$, be such that $r\equiv n \pmod {\rho }$. The main result of this note says that if $(k-\ell)/\ell$ is small in terms of $\rho$, then the number of $(k,\ell)$-sum-free subsets of $[1,n]$ is equal to $(\varphi(\rho)+\varphi_r(\rho)+o(1)) 2^{\lfloor n/\rho \rfloor}$, where $\varphi_r(x)$ denotes the number of positive integers $m\le r$ relatively prime to $x$ and $\varphi(x)=\varphi_x(x)$.

scholarly journals ON THE WARING–GOLDBACH PROBLEM FOR CUBES

2009 ◽  
Vol 51 (3) ◽  
pp. 703-712 ◽  
Author(s):  
JÖRG BRÜDERN ◽  
KOICHI KAWADA
Keyword(s):  
Natural Number ◽  
Large Integer ◽  
Natural Numbers ◽  
Goldbach Problem ◽  
Almost All

AbstractWe prove that almost all natural numbers satisfying certain necessary congruence conditions can be written as the sum of two cubes of primes and two cubes of P2-numbers, where, as usual, we call a natural number a P2-number when it is a prime or the product of two primes. From this result we also deduce that every sufficiently large integer can be written as the sum of eight cubes of P2-numbers.

Exceptional Set in Waring–Goldbach Problem Involving Squares, Cubes and Sixth Powers

2021 ◽  
Vol 58 (1) ◽  
pp. 84-103
Author(s):  
Jinjiang Li ◽  
Min Zhang ◽  
Haonan Zhao
Keyword(s):  
Prime Numbers ◽  
Large Integer ◽  
Exceptional Set ◽  
Goldbach Problem ◽  
Positive Integers

Let N be a sufficiently large integer. In this paper, it is proved that, with at most O(N 119/270+s) exceptions, all even positive integers up to N can be represented in the form where p1, p2, p3, p4, p5, p6 are prime numbers.

scholarly journals On the Number of Products Which Form Perfect Powers and Discriminants of Multiquadratic Extensions

2019 ◽  
Author(s):  
Régis de la Bretèche ◽  
Pär Kurlberg ◽  
Igor E Shparlinski
Keyword(s):  
Asymptotic Formula ◽  
Number Fields ◽  
Bounded Size ◽  
Positive Integers

Abstract We study some counting questions concerning products of positive integers $u_1, \ldots , u_n$, which form a nonzero perfect square, or more generally, a perfect $k$-th power. We obtain an asymptotic formula for the number of such integers of bounded size and in particular improve and generalize a result of D. I. Tolev (2011). We also use similar ideas to count the discriminants of number fields that are multiquadratic extensions of ${\mathbb{Q}}$ and improve and generalize a result of N. Rome (2017).

Note on the representation of integers as sums of certain powers

1969 ◽  
Vol 65 (2) ◽  
pp. 445-446 ◽  
Author(s):  
K. Thanigasalam
Keyword(s):  
Asymptotic Formula ◽  
Natural Number ◽  
Natural Numbers ◽  
Waring’S Problem ◽  
Waring's Problem ◽  
Representation Of Integers ◽  
Image Position ◽  
Large Natural Number

In the paper entitled ‘Asymptotic formula in a generalized Waring's problem’, I established an asymptotic formula for the number of representations of a large natural number N in the formwhere x1, x2, …, x7 and k are natural numbers with k ≥ 4 (see (2) Theorem 2).

Sign in / Sign up

Export Citation Format

Share Document