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  • Print publication year: 1997
  • Online publication date: April 2010

4 - Goldbach Numbers and Uniform Distribution

Summary

Introduction As an illustrative example of their celebrated circle method, Hardy and Littlewood were able to show that subject to the truth of the Generalized Riemann Hypothesis, almost all even natural numbers are the sum of two primes, the yet unproven hypothesis being removed later as a consequence of Vinogradov's work. Natural numbers which are representable as the sum of two primes are called Goldbach numbers, and it is still not known whether all, or at least all but finitely many, even positive integers ≥ 4 are of this form. The best estimate for the number of possible exceptions is due to Montgomery and Vaughan [4]. They showed that all but O(X1–δ) even natural numbers not exceeding X are Goldbach numbers, for some small δ > 0.

More information about possible exceptions can be obtained by considering thin subsequences of the even numbers, with the aim of showing that almost all numbers in the subsequence are Goldbach numbers. In this direction, short intervals have been treated by various authors. It is now known that almost all even numbers in the interval [X, X + X11/160+ε] are Goldbach numbers (see Baker, Harman and Pintz [1]). Perelli [5] has shown that almost all even positive values of an integer polynomial satisfying some natural arithmetical conditions are Goldbach numbers.

In this paper we give further examples of sequences with this property. They arise, roughly speaking, as integer approximations to values of real-valued functions at integers points whose fractional parts are uniformly distributed modulo one. We need some notation to make this precise.