The quadratic Waring-Goldbach problem

It is conjectured that Lagrange's theorem of four squares is true for prime variables, i.e. all positive integers n with are the sum of four squares of primes. In this paper, the size for the exceptional set in the above conjecture is reduced to .     

On a Waring-Goldbach-type problem for fourth powers

In this paper, we prove that every sufficiently large positive integer satisfying some necessary congruence conditions can be represented by the sum of a fourth power of integer and twelve fourth powers of prime numbers.     

Two results on powers of 2 in Waring-Goldbach problem

In this paper, it is proved that every sufficiently large odd integer is a sum of a prime, four cubes of primes and 106 powers of 2. What is more, every sufficiently large even integer is a sum of two squares of primes, four cubes of primes and 211 powers of 2.     

A note on the Waring-Goldbach problem

The focus of this paper will be the extension of the Waring-Goldbach problem to all sufficiently large integers, without congruence restrictions. By reintroducing the effect of small primes, we are able to consider questions which more naturally resemble Waring's problem and the Goldbach conjecture. We extend the results of S.S. Pillai by considering the problem without the use of zero as an addend and we give a small improvement on the number of additional terms required.     

On the number of representations in the ternary Goldbach problem with one prime number in a given residue class

For

     

On linear equations with prime variables of special type

Let Pk denote any integer with no more than k prime factors, counted according to multiplicity. It is proved that for every sufficiently large odd integer , the equation p₁+p₂+p₃=n is solvable in prime variables p₁,p₂,p₃ such that p₁+2=P₂, , and for almost all sufficiently large even integer , the equation p₁+p₂=n is solvable in prime variables p₁,p₂ such that p₁+2=P₂.     

On sums of a prime and four prime squares in short intervals

In this paper, we are able to sharpen Hua's classical result by showing that each sufficiently large integer can be written as

     

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 p ₁ + p ₂ = n is solvable in prime variables p ₁, p ₂ such that p ₁ + 2 = P ₃, and for every sufficiently large odd integer satisfying ≢ 1(mod 6), the equation p ₁ + p ₂ + p ₃ = is solvable in prime variables p ₁, p ₂, p ₃ such that p ₁ + 2 = P ₂, p ₂ + 2 = P ₃. Here P _k denotes any integer with no more than k prime factors, counted according to multiplicity.     

On exponential sums over primes in short intervals

Let Λ(n) be the von Mangoldt function, x real and y small compared with x. This paper gives a non-trivial estimate on the exponential sum over primes in short intervals for all α ∈ [0,1] whenever . This result is as good as what was previously derived from the Generalized Riemann Hypothesis.     

An application of algebraic sieve theory

Let K be a fixed totally real algebraic number field of finite degree over the rationals. The theme of this paper is the problem about the occurrence of algebraic almost-primes in a polynomial sequence generated by an irreducible polynomial of K with prime arguments. The method is based on a weighted upper and lower linear Selberg-type sieve in K and makes use of a multidimensional algebraic version of Bombieris theorem on primes in arithmetic progressions.     

On the exceptional set in Goldbach’s problem in short intervals

Let 1/5 < θ ≤ 1. We prove that there exists a positive constant δ such that the number of even integers in the interval [X, X + X ^θ] which are not a sum of two primes is 《 X ^θ−δ. The proof uses the circle method, a sieve method, exponential sum estimates and zero-density estimates for L-functions. Current address: Department of Mathematics, 20014 University of Turku, Finland. Author’s address: Department of Mathematics, University of London, Royal Holloway, Egham, Surrey TW20 0EX, UK     

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. (Received 11 January 2000; in revised form 4 December 2000)     

Additive functions on arithmetic progressions with large moduli

In this paper we study additive functions on arithmetic progressions with large moduli. We are able to improve some former results given by Elliott.     

On sums of squares of primes II

In this paper we continue our study, begun in G. Harman and A.V. Kumchev (2006) [10], of the exceptional set of integers, not restricted by elementary congruence conditions, which cannot be represented as sums of three or four squares of primes. We correct a serious oversight in our first paper, but make further progress on the exponential sums estimates needed, together with an embellishment of the previous sieve technique employed. This leads to an improvement in our bounds for the maximal size of the exceptional sets.     

An Additive Problem with Piatetski-Shapiro Primes and Almost-Primes

Suppose that . We prove a theorem of Bombieri-Vinogradov type for the Piatetski-Shapiro primes p = [n ^1/ and show that every sufficiently large even integer can be written as a sum of a Piatetski-Shapiro prime and an almost-prime.Received November 29, 2001; in revised form August 21, 2002 Published online October 15, 2003     

Squares of Primes and Powers of 2

 As an extension of the Linnik-Gallagher results on the “almost Goldbach” problem, we prove, among other things, that there exists a positive integer k ₀ such that every large even integer is a sum of four squares of primes and k ₀ powers of 2. (Received 7 September 1998; in revised form 3 May 1999)     

On sums of three integers with a fixed number of prime factors

Let N be sufficiently large odd integer. It is proved that the equation N=n₁+n₂+n₃ has solutions, where n_i has a fixed number of prime factors, and an asymptotic formula holds for the number of representations.     

On the fourth moment in the Rankin-Selberg problem

If

On Bombieri's asymptotic sieve

We improve the error term in the Bombieri asymptotic sieve when the summation is restricted to integers having at most two prime factors. This results in a refined bilinear decomposition for the characteristic function of the primes that enables us to get a best possible estimate for the trigonometric polynomial over primes.