Refinements of Goldbach's conjecture,and the generalized Riemann hypothesis

Abstract

We present three remarks on Goldbach's problem. First we suggest a refinement of Hardy and Littlewood's conjecture for the number of representations of $2n$ as the sum of two primes positing an estimate with a very small error term. Next we show that if a strong form of Goldbach's conjecture is true then every even integer is the sum of two primes from a rather sparse set of primes. Finally we show that an averaged strong form of Goldbach's conjecture is equivalent to the Generalized Riemann Hypothesis; as well as a similar equivalence to estimates for the number of ways of writing integers as the sum of $k$ primes.