A remarkable property of the “singular series” in waring’s problem and its relation to hypothesis K of hardy and littlewood

A classical conjecture in the additive theory of numbers is that all sufficiently large natural numbers may be written as the sum of four positive cubes of integers. This is known as the Four Cubes Problem, and since the pioneering work of Hardy and Littlewood one expects a much more precise quantitative form of the conjecture to hold. Let v(n) be the number of representations of n in the proposed manner. Then the expected formula takes the shapewhere (n) is the singular series associated with four cubes as familiar in the Hardy–Littlewood theory.

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Some problems of ‘Partitio Numerorum’: IV. The singular series in Waring’s Problem and the value of the number G (k)

Festschrift David Hilbert zu Seinem Sechzigsten Geburtstag am 23. Januar 1922 ◽

10.1007/978-3-642-52605-3_39 ◽

1922 ◽

pp. 365-392

Author(s):

G. H. Hardy ◽

J. E. Littlewood

Keyword(s):

Singular Series ◽

Waring’S Problem ◽

Waring's Problem ◽

Partitio Numerorum

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Some problems of ‘Partitio Numerorum’: IV. The singular series in Waring’s Problem and the value of the number G (k)

Festschrift ◽

10.1007/978-3-642-61810-9_39 ◽

1982 ◽

pp. 365-392

Author(s):

G. H. Hardy ◽

J. E. Littlewood

Keyword(s):

Singular Series ◽

Waring’S Problem ◽

Waring's Problem ◽

Partitio Numerorum

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Arithmetic of higher-dimensional orbifolds and a mixed Waring problem

Mathematische Zeitschrift ◽

10.1007/s00209-021-02695-w ◽

2021 ◽

Author(s):

Tim Browning ◽

Shuntaro Yamagishi

Keyword(s):

Circle Method ◽

Rational Points ◽

Mean Value ◽

Mean Value Theorem ◽

Waring’S Problem ◽

Waring Problem ◽

Waring's Problem ◽

Main Conjecture ◽

Higher Dimensional ◽

Thin Set

AbstractWe study the density of rational points on a higher-dimensional orbifold $$(\mathbb {P}^{n-1},\Delta )$$ ( P n - 1 , Δ ) when $$\Delta $$ Δ is a $$\mathbb {Q}$$ Q -divisor involving hyperplanes. This allows us to address a question of Tanimoto about whether the set of rational points on such an orbifold constitutes a thin set. Our approach relies on the Hardy–Littlewood circle method to first study an asymptotic version of Waring’s problem for mixed powers. In doing so we make crucial use of the recent resolution of the main conjecture in Vinogradov’s mean value theorem, due to Bourgain–Demeter–Guth and Wooley.

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

Nagoya Mathematical Journal ◽

10.1017/s0027763000007893 ◽

2001 ◽

Vol 163 ◽

pp. 13-53 ◽

Cited By ~ 4

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.

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