Functiones et Approximatio Commentarii Mathematici

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

Andrew Granville

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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.

Article information

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

Dates
First available in Project Euclid: 18 December 2008

Permanent link to this document
https://projecteuclid.org/euclid.facm/1229618748

Digital Object Identifier
doi:10.7169/facm/1229618748

Mathematical Reviews number (MathSciNet)
MR2357316

Zentralblatt MATH identifier
1230.11123

Subjects
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

Keywords
Goldbach additive number theory Riemann zeta function

Citation

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.


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