Functiones et Approximatio Commentarii Mathematici
- Funct. Approx. Comment. Math.
- Volume 37, Number 1 (2007), 159-173.
Refinements of Goldbach's conjecture,and the generalized Riemann hypothesis
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.
Funct. Approx. Comment. Math., Volume 37, Number 1 (2007), 159-173.
First available in Project Euclid: 18 December 2008
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Granville, Andrew. Refinements of Goldbach's conjecture,and the generalized Riemann hypothesis. Funct. Approx. Comment. Math. 37 (2007), no. 1, 159--173. doi:10.7169/facm/1229618748. https://projecteuclid.org/euclid.facm/1229618748