scholarly journals The asymptotic formula in Waring’s problem: Higher order expansions

2018 ◽  
Vol 2018 (742) ◽  
pp. 17-46
Author(s):  
Robert C. Vaughan ◽  
Trevor D. Wooley
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Formula ◽  
Natural Number ◽  
Higher Order ◽  
Waring’S Problem ◽  
Waring's Problem ◽  
Large Natural Number

Abstract When {k>1} and s is sufficiently large in terms of k, we derive an explicit multi-term asymptotic expansion for the number of representations of a large natural number as the sum of s positive integral k-th powers.

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).

scholarly journals A light-weight version of Waring's problem

2004 ◽  
Vol 76 (3) ◽  
pp. 303-316 ◽  
Author(s):  
Trevor D. Wooley
Keyword(s):  
Asymptotic Formula ◽  
Large Integer ◽  
Light Weight ◽  
Natural Numbers ◽  
Waring’S Problem ◽  
Waring's Problem ◽  
Large Solutions ◽  
Homogeneous Weight

AbstractAn asymptotic formula is established for the number of representations of a large integer as the sum of kth powers of natural numbers, in which each representation is counted with a homogeneous weight that de-emphasises the large solutions. Such an asymptotic formula necessarily fails when this weight is excessively light.

Asymptotic formula in a generalized Waring's problem

1967 ◽  
Vol 63 (1) ◽  
pp. 87-98 ◽  
Author(s):  
K. Thanigasalam
Keyword(s):  
Asymptotic Formula ◽  
Waring’S Problem ◽  
Waring's Problem

Introduction. In the classical Waring's problem, the following asymptotic formula was established by Vinogradov (see Vinogradov (2), Chapter VII).

On Waring's problem for cubes and biquadrates. II

1988 ◽  
Vol 104 (2) ◽  
pp. 199-206 ◽  
Author(s):  
Jörg Brüdern
Keyword(s):  
Lower Bound ◽  
Asymptotic Formula ◽  
Riemann Hypothesis ◽  
Waring’S Problem ◽  
Waring's Problem ◽  
Asymptotic Formulae ◽  
Sixth Moment ◽  
Almost All

In discussing the consequences of a conditional estimate for the sixth moment of cubic Weyl sums, Hooley [4] established asymptotic formulae for the number ν(n) of representations of n as the sum of a square and five cubes, and for ν(n), defined similarly with six cubes and two biquadrates. The condition here is the truth of the Riemann Hypothesis for a certain Hasse–Weil L-function. Recently Vaughan [8] has shown unconditionally , a lower bound of the size suggested by the conditional asymptotic formula. In the corresponding problem for ν(n) the author [1] was able to deduce ν(n) > 0, as a by-product of the result that almost all numbers can be expressed as the sum of three cubes and one biquadrate. As promised in the first paper of this series we return to the problem of bounding ν(n) from below.

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