scholarly journals Generalising the Hardy–Littlewood method for primes

2009 ◽  
pp. 373-399 ◽  
Author(s):  
Ben Green
Keyword(s):  
Hardy Littlewood Method

scholarly journals On Waring’s problem: Three cubes and a sixth power

2001 ◽  
Vol 163 ◽  
pp. 13-53 ◽  
Author(s):  
Jörg Brüdern ◽  
Trevor D. Wooley
Keyword(s):  
Exponential Sums ◽  
Large Integer ◽  
Natural Numbers ◽  
Waring’S Problem ◽  
Smooth Numbers ◽  
Waring's Problem ◽  
Order Of Magnitude ◽  
Almost All ◽  
Hardy Littlewood Method

We establish that almost all natural numbers not congruent to 5 modulo 9 are the sum of three cubes and a sixth power of natural numbers, and show, moreover, that the number of such representations is almost always of the expected order of magnitude. As a corollary, the number of representations of a large integer as the sum of six cubes and two sixth powers has the expected order of magnitude. Our results depend on a certain seventh moment of cubic Weyl sums restricted to minor arcs, the latest developments in the theory of exponential sums over smooth numbers, and recent technology for controlling the major arcs in the Hardy-Littlewood method, together with the use of a novel quasi-smooth set of integers.

scholarly journals Rational points on linear slices of diagonal hypersurfaces

2015 ◽  
Vol 218 ◽  
pp. 51-100
Author(s):  
Jörg Brüdern ◽  
Olivier Robert
Keyword(s):  
Asymptotic Formula ◽  
Rational Points ◽  
Independent Interest ◽  
The Way ◽  
Hardy Littlewood Method

AbstractAn asymptotic formula is obtained for the number of rational points of bounded height on the class of varieties described in the title line. The formula is proved via the Hardy-Littlewood method, and along the way we establish two new results on Weyl sums that are of some independent interest.

scholarly journals An extension of Waring's problems

1933 ◽  
Vol 232 (707-720) ◽  
pp. 1-26 ◽  
Keyword(s):  
The Other ◽  
Waring’S Problem ◽  
Waring's Problem ◽  
Large N ◽  
Hardy Littlewood Method

An interesting extension of Waring’s famous problem is the following:— Can every sufficiently large n be expressed, as the sum of s almost equal k-th powers; or, more generally, can every sufficiently large n be expressed as the sum of s positive k-th powers almost proportional to s arbitrarily assigned positive numbers Xl5 x2, ... x, ? I have developed two methods to discuss this problem, one based on the Hardy-Littlewood method for the solution of WARING’s problem and the other on the new VINOGRADOFF method* for the solution of the same problem. In this paper I shall discuss the case k3 by the first of these methods. The case of five or more squares may be treated in the same way, and the results are similar; somewhat deeper and more troublesome analysis is required to deal with the case of four squares. The principle of the method employed here is that of weighting ” the various representations of n as the sum of s k-th powers in such a way as to make predominant the particular representation of which we are in search.

Six Primes and an Almost Prime in Four Linear Equations

1998 ◽  
Vol 50 (3) ◽  
pp. 465-486 ◽  
Author(s):  
Antal Balog
Keyword(s):  
Asymptotic Formula ◽  
Linear Equations ◽  
Arithmetic Mean ◽  
Sieve Method ◽  
Arithmetic Means ◽  
Almost Prime ◽  
Hardy Littlewood Method

AbstractThere are infinitely many triplets of primes p, q, r such that the arithmetic means of any two of them, are also primes. We give an asymptotic formula for the number of such triplets up to a limit. The more involved problem of asking that in addition to the above the arithmetic mean of all three of them, is also prime seems to be out of reach. We show by combining the Hardy-Littlewood method with the sieve method that there are quite a few triplets for which six of the seven entries are primes and the last is almost prime.

scholarly journals Exceptional Set for Sums of Symmetric Mixed Powers of Primes

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 367
Author(s):  
Jinjiang Li ◽  
Chao Liu ◽  
Zhuo Zhang ◽  
Min Zhang
Keyword(s):  
Upper Bound ◽  
Exceptional Set ◽  
Upper Bound Estimate ◽  
Hardy Littlewood Method

The main purpose of this paper is to use the Hardy–Littlewood method to study the solvability of mixed powers of primes. To be specific, we consider the even integers represented as the sum of one prime, one square of prime, one cube of prime, and one biquadrate of prime. However, this representation can not be realized for all even integers. In this paper, we establish the exceptional set of this kind of representation and give an upper bound estimate.

Cubic forms in thirty-two variables

1959 ◽  
Vol 251 (993) ◽  
pp. 193-232 ◽  
Keyword(s):  
Exponential Sum ◽  
Definite Integral ◽  
Cubic Form ◽  
New Form ◽  
Cubic Forms ◽  
Hardy Littlewood Method

It is proved that if C(xu...,*„) is any cubic form in n variables, with integral coefficients, then the equation C{xu ...,*„) = 0 has a solution in integers xXi...,xn, not all 0, provided n is at least 32. The proof is based on the Hardy-Littlewood method, involving the dissection into parts of a definite integral, but new principles are needed for estimating an exponential sum containing a general cubic form. The estimates obtained here are conditional on the form not splitting in a particular manner; when it does so split, the same treatment is applied to the new form, and ultimately the proof is made to depend on known results.

scholarly journals Rational points on linear slices of diagonal hypersurfaces

2015 ◽  
Vol 218 ◽  
pp. 51-100 ◽  
Author(s):  
Jörg Brüdern ◽  
Olivier Robert
Keyword(s):  
Asymptotic Formula ◽  
Rational Points ◽  
Independent Interest ◽  
The Way ◽  
Hardy Littlewood Method

AbstractAn asymptotic formula is obtained for the number of rational points of bounded height on the class of varieties described in the title line. The formula is proved via the Hardy-Littlewood method, and along the way we establish two new results on Weyl sums that are of some independent interest.

A Divisor Problem for Values of Polynomials

1992 ◽  
Vol 35 (1) ◽  
pp. 108-115
Author(s):  
Armel Mercier ◽  
Werner Georg Nowak
Keyword(s):  
Exponential Sums ◽  
Modern Method ◽  
Divisor Problem ◽  
Arithmetical Function ◽  
Asymptotic Result ◽  
Average Order ◽  
Equal Degree ◽  
Image Position ◽  
Classical Divisor ◽  
Hardy Littlewood Method

AbstractIn this article we investigate the average order of the arithmetical functionwhere p1(t), p2(t) are polynomials in Z [t], of equal degree, positive and increasing for t ≥ 1. Using the modern method for the estimation of exponential sums ("Discrete Hardy-Littlewood Method"), we establish an asymptotic result which is as sharp as the best one known for the classical divisor problem.

Sign in / Sign up

Export Citation Format

Share Document