Functiones et Approximatio Commentarii Mathematici

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

Andrew Granville

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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.

Article information

Funct. Approx. Comment. Math., Volume 37, Number 1 (2007), 159-173.

First available in Project Euclid: 18 December 2008

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Zentralblatt MATH identifier

Primary: 11P32: Goldbach-type theorems; other additive questions involving primes
Secondary: 11M26: Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses

Goldbach additive number theory Riemann zeta function


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.

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