On Siegel zeros of Hecke-Landau zeta-functions

The primary concern of this paper is to deal with Siegel zeros of Hecke-Landau zeta-functions in an algebraic number field of finite degree over the rationals. As in the rational case with DirichletL-functions, the location of such zeros is closely connected with lower bounds for the corresponding zeta-functions at the points=1. This will be the theme in the first part of the paper. In this second part we first derive a form of the Brun-Titchmarsh theorem in the setting of a number field which is appropriate in our context. Then we turn our attention to the fact that an improvement of the constant in this inequality would lead to the nonexistence of Siegel zeros. The procedure is based on a weighted algebraic form of Selberg's upper bound sieve.     

Goldbach problem in polynomial values

An even integern≥6 is called a Goldbach number if it is the sum of two odd primes. The Goldbach conjecture says that every even numbern≤6 is a Goldbach number. In this paper, we study the mean-square upper bound of the error term related to the Goldbach problem, in polynomial values over short intervals, uniformly with respect to the height of a polynomial of fixed degree.     

The major arcs in the ternary Goldbach problem

We enlarge considerably the major arcs in the ternary Goldbach problem. This revised version was published online in June 2006 with corrections to the Cover Date.     

The general Goldbach problem with Beatty primes

In this paper, we are concerned with the representation of large positive integers as the sum of a fixed number of prime numbers taken from Beatty sequences. We are able to avoid any conditions on the Beatty sequences that they be of finite type which have been required hitherto. We use a form of the Hardy–Littlewood–Vinogradov method, but the proofs are quite delicate.     

On Goldbach problem concerning factorization of meromorphic functions

The authors prove in this paper the following

Theorem  

Letf (Z) be any transcendental meromorphic function in the plane. Then for any given non-constant fractional linear function (Z), the set

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 [`(n)]{\bar n} satisfying [`(n)]{\bar n} ≢ 1(mod 6), the equation p ₁ + p ₂ + p ₃ = [`(n)]{\bar n} 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.     

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.     

The ternary Goldbach problem with a prime and two isolated primes

We prove that under the assumption of the Generalized Riemann Hypothesis each sufficiently large odd integer can be expressed as the sum of a prime and two isolated primes.     

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     

The ternary Goldbach problem with arithmetic weights attached to two of the variables

We consider the ternary Goldbach problem with two prime variables of the form k²+m²+1 and find an asymptotic formula for the number of its solutions.     

Inverse M-matrix completion problem with zeros in the inverse completion

In this paper, we obtain an inverse M-matrix completion, with zeros in the inverse completion, of a noncombinatorially symmetric partial inverse M-matrix, when the associated graph is acyclic without specified paths or, in the other case, when the subgraph induced by the vertices of any cycle or specified path is a clique.     

Inverse problem for zeros of certain koebe-related functions

If w₁,…,w _N is a finite sequence of nonzero points in the unit disk, then there are distinct points λ₁,…, λ_N on the unit circle and positive numbers Μ₁,…,Μ _N such that is the zero sequence of the function 1 — . The points λ₁,…, λ_N and numbers Μ₁,…,Μ_N are unique (except for reorderings).     

Trigonometric polynomials with many real zeros and a Littlewood-type problem

We examine the size of a real trigonometric polynomial of degree at most having at least zeros in (counting multiplicities). This result is then used to give a new proof of a theorem of Littlewood concerning flatness of unimodular trigonometric polynomials. Our proof is shorter and simpler than Littlewood's. Moreover our constant is explicit in contrast to Littlewood's approach, which is indirect.

     

Right-angled Coxeter polytopes, hyperbolic six-manifolds, and a problem of Siegel

By gluing together the sides of eight copies of an all-right angled hyperbolic six-dimensional polytope, two orientable hyperbolic six-manifolds with Euler characteristic ?1 are constructed. They are the first known examples of orientable hyperbolic six-manifolds having the smallest possible volume.     

On the second Goldbach conjecture

The results in this paper suggest that Goldbach's conjecture that every odd number is the sum of three primes is true even under the requirement that two of the primes be the same and the third be arbitrarily small.     

Fixed zeros in the model matching problem for systems over semirings

This paper studies “fixed zeros” of solutions to the model matching problem for systems over semirings. Such systems have been used to model queueing systems, communication networks, and manufacturing systems. The main contribution of this paper is the discovery of two fixed zero structures, which possess a connection with the extended zero semimodules of solutions to the model matching problem. Intuitively, the fixed zeros provides an essential component that is obtained from the solutions to the model matching problem. For discrete event dynamic systems modeled in max-plus algebra, a common Petri net component constructed from the solutions to the model matching problem can be discovered from the fixed zero structure.     

On checking the Goldbach conjecture

This paper describes an attempt to check Goldbach's conjecture on a digital computer. The validity of the conjecture has been numerically verified up toN=33,000,000.