Binary Goldbach Problem Research Articles

Density versions of the binary Goldbach problem

Density versions of the binary Goldbach problem

On Linnik’s approximation to Goldbach’s problem. II

Linnik considered about 70 years ago the following approximation to the binary Goldbach problem. Is it possible to give a fixed integer K such that every sufficiently large even integer could be written as the sum of two primes and K powers of two? He solved the problem affirmatively with an unspecified large K. The first explicit result $$(K=54\,000)$$ appeared at the end of the 1990's. In the present work we show that $$K=8$$ powers of two suffices without supposing any unproved hypothesis.

Large sieve inequality with sparse sets of moduli applied to Goldbach conjecture

We use the large sieve inequality with sparse sets of moduli to prove a new estimate for exponential sums over primes. Subsequently, we apply this estimate to establish new results on the binary Goldbach problem where the primes are restricted to given arithmetic progressions.

The binary Goldbach problem with arithmetic weights attached to one of the variables

We study the binary Goldbach problem with arithmetic weights attached to one of the variables.

Rank-one quadratic twists of an infinite family of elliptic curves

A conjecture of Goldfeld implies that a positive proportion of quadratic twists of an elliptic curve E /ℚ has (analytic) rank 1. This assertion has been confirmed by Vatsal [Math. Ann. 311: 791–794, 1998] and the first author [Acta Arith. 114: 391–396, 2004] for only two elliptic curves. Here we confirm this assertion for infinitely many elliptic curves E /ℚ using the Heegner divisors, the 3-part of the class groups of quadratic fields, and a variant of the binary Goldbach problem for polynomials.

Additive Representation in Thin Sequences, VII: Restricted Moments of the Number of Representations

Let $R_{s,k}(n)$ denote the number of representations of $n$ as the sum of $s$ positive integral $k$th powers. We develop methods to establish asymptotic formulae for moments of $R_{s,k}(n)$ in which $n$ is restricted to a thin sequence consisting of values of a given polynomial. As a first example, we discuss the case of cubes, and present conclusions for sums of 6 and 7 cubes in which the polynomial is quadratic, and for sums of 7 cubes in which the polynomial is cubic. We also consider the case wherein the exponent $k$ is large, and briefly describe corresponding results for the binary Goldbach problem.

A mean value theorem on the binary Goldbach problem and its application

We study the binary Goldbach problem with one prime number in a given residue class, and obtain a mean value theorem. As an application, we prove that for almost all sufficiently large even integers n satisfying n ≢ 2(mod 6), the equation p1 + p2 = n is solvable in prime variables p1, p2 such that p1 + 2 = P3, and for every sufficiently large odd integer \({\bar n}\) satisfying \({\bar n}\) ≢ 1(mod 6), the equation p1 + p2 + p3 = \({\bar n}\) is solvable in prime variables p1, p2, p3 such that p1 + 2 = P2, p2 + 2 = P3. Here Pk denotes any integer with no more than k prime factors, counted according to multiplicity.

The binary Goldbach problem with one prime of the form [formula omitted

We prove that almost all integers n ≡ 0 or 4 (mod 6) can be written in the form n = p 1 + p 2 , where p 1 = k 2 + l 2 + 1 with ( k , l ) = 1 . The proof is an application of the half-dimensional and linear sieves with arithmetic information coming from the circle method and the Bombieri–Vinogradov prime number theorem.

Additive representation in thin sequences, V: Mixed problems of Waring's type

In the first part of this series of papers (see Brudern, Kawada and Wooley [2]), we introduced an approach to additive problems in which one seeks to establish that almost all natural numbers in some fixed polynomial sequence are represented in a prescribed manner, thereby deriving non-trivial estimates for exceptional sets in thin sequences. We illustrated our methods by obtaining upper bounds for the exceptional sets associated with the representation of integers from quadratic, or cubic, polynomial sequences by sums of six cubes of positive integers. In subsequent parts of the series (see Brudern, Kawada and Wooley [3], [4], [5]), we adapted our core methods so as to tackle problems associated with the binary Goldbach problem, the expected asymptotic formula for the number of representations, and lower bounds for the number of integers represented in some prescribed manner. As is apparent from the opening part of this series, our methods are of great flexibility. The aim of the present paper is to provide variants of the ideas developed in the preceding opera, and here we will be concerned solely with methods which provide estimates for the size of exceptional sets in representation problems. The discerning reader will recognise that in several of the more exotic problems mentioned below, it is the existence of a non-trivial estimate for the exceptional set in question which is of interest. The investigation of the sharpest attainable estimate for this exceptional set should be politely deferred beyond any future occasion. We begin by exploring exceptional sets in polynomial sequences for additive problems involving mixed powers. Here one finds that sharp mean value estimates for mixed sums of powers, familiar to aficionados of the circle method, lead to surprisingly strong conclusions. Our first results, which we establish in §2, involve problems containing a block of four cubes. Here and elsewhere, ∗ Packard Fellow, and supported in part by NSF grant DMS-9622773. This paper benefitted from visits of various of the authors to Ann Arbor, Kyoto, Oberwolfach and Stuttgart, and the authors collectively thank these institutions for their hospitality and excellent working conditions. Received August 16, 2000.

Some Applications of a Bombieri-Type Theoremin Totally Real Number Fields

The primary concern of this paper is to present three further applications of a multi-dimensional version of Bombieri’s theorem on primes in arithmetic progressions in the setting of a totally real algebraic number field K. First, we deal with the order of magnitude of a greatest (relative to its norm) prime ideal factor of \(\), where the product runs over prime arguments ω of a given irreducible polynomial F which lie in a certain lattice point region. Then, we turn our attention to the problem about the occurrence of algebraic primes in a polynomial sequence generated by an irreducible polynomial of K with prime arguments. Finally, we give further contributions to the binary Goldbach problem in K.

Additive representation in thin sequences, II: The binary Goldbach problem

Additive representation in thin sequences, II: The binary Goldbach problem

On exponential sums over primes in arithmetic progressions

with a wide uniformity in real a, where A is the von Mangoldt function and, for real 6, e{6) = exp(2niO). By using a combinatrial identity, R. C. Vaughan presented an elegant simple argument on it, see [2], for instance. J.-r.Chen's theorem on the binary Goldbach problem is built upon the linear sieve and the mean prime number theorem, vide [5]. According to H. Iwaniec [6], the Rosser's weight of the linear sieve has the property. An arithmetic function k is called well-factorable of level D, if for any D\,Di > 1, D = D\Dj, there exist two functions k\and ki supporting in (0,Z>i] and (0,Z>2] respectively such that |^i| < 1, \ki\< 1 and k = k\* kj. Also the mean prime number theorem has been surprisingly developed by E. Fouvry and H. Iwaniec [4],E. Fouvry [3] and E. Bombieri, J.-B. Friedlander and H. Iwaniec [1].In [1] they established a non-trivial bound of the averaging sum

Some remarks on the binary Goldbach problem

Some remarks on the binary Goldbach problem