Exceptional Set for Sums of Symmetric Mixed Powers of Primes

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.

A problem considered by Friedlander & Iwaniec and the discrete Hardy-Littlewood method

Following Friedlander & Iwaniec [FRIEDLANDER, J. B.—IWANIEC, H.:

Rational points on linear slices of diagonal hypersurfaces

An 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.

Rational points on linear slices of diagonal hypersurfaces

An 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.

Points de hauteur bornée sur les hypersurfaces lisses de l'espace triprojectif

Nous démontrons ici la conjecture de Batyrev et Manin pour le nombre de points de hauteur bornée de certaines hypersurfaces de l'espace triprojectif de tridegré (1, 1, 1). La constante intervenant dans le résultat final est celle conjecturée par Peyre. La méthode utilisée est inspirée de celle développée par Schindler pour traiter le cas des hypersurfaces des espaces biprojectifs. Celle-ci est essentiellement basée sur la méthode du cercle de Hardy–Littlewood. We prove the Batyrev–Manin conjecture for the number of points of bounded height on some smooth hypersurfaces of the triprojective space of tridegree (1, 1, 1). The constant appearing in the final result is the one conjectured by Peyre. The method used is the one developed by Schindler to study the case of hypersurfaces of biprojective spaces. It is essentially based on the Hardy–Littlewood method.

Slim Exceptional Sets for Sums of Cubes

We investigate exceptional sets associated with various additive problems involving sums of cubes. By developing a method wherein an exponential sum over the set of exceptions is employed explicitly within the Hardy-Littlewood method, we are better able to exploit excess variables. By way of illustration, we show that the number of odd integers not divisible by 9, and not exceeding X, that fail to have a representation as the sum of 7 cubes of prime numbers, is O(X23/36+ε). For sums of eight cubes of prime numbers, the corresponding number of exceptional integers is O(X11/36+ε).

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

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.

Six Primes and an Almost Prime in Four Linear Equations

There 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.