## Functiones et Approximatio Commentarii Mathematici

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

Andrew Granville

#### 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

https://projecteuclid.org/euclid.facm/1229618748

Digital Object Identifier
doi:10.7169/facm/1229618748

Mathematical Reviews number (MathSciNet)
MR2357316

Zentralblatt MATH identifier
1230.11123

#### 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.

#### References

• Noga Alon and Joel H. Spencer, The Probabilistic Method, Wiley 1992.
• N. Chudakov, On Goldbach's problem (Russian), Dokl. Akad. Nauk SSSR 17 (1937), 331--334.
• Jean-Marc Deshouillers, Andrew Granville, Wladyslaw Narkiewicz and Carl Pomerance, An upper bound in Goldbach's problem, Math. Comp. 61 (1993), 209--213.
• T. Estermann, On Goldbach's problem: Proof that almost all even positive integers are sums of two primes, Proc. London Math. Soc 44 (1938), 307--314.
• G.H. Hardy and J.E. Littlewood, Some problems of 'partitio numerorum'; V: A further contribution to the study of Goldbach's problem Proc. London Math. Soc 22 (1924), 46--56.
• M. Kolountzakis, On the additive complements of the primes and sets of similar growth, Acta Arith 77 (1996), 1--8.
• J.E. Littlewood, Distribution des nombres premiers, C. R. Acad. Sci. Paris 158 (1914), 1869--1872.
• H.L. Montgomery and R.C. Vaughan, Error terms in additive prime number theory and the GRH, to appear.
• H.L. Montgomery and R.C. Vaughan, The exceptional set in Goldbach's problem, Acta. Arith 27 (1975), 353--370.
• J. Pintz, Explicit formulas and the exceptional set in Goldbach's problem, to appear.
• O. Ramaré, On S'nirel'man's constant, Ann. Sci. Norm. Super. Pisa 21 (1995), 645--705.
• J. Richstein, Verifying the Goldbach conjecture up to $4\cdot 10^14$, Math. Comp 70 (2000), 1745--1749.
• J. G. Van der Corput, Sur l'hypothèse de Goldbach, Proc. Acad. Wet. Amsterdam 41 (1938), 76--80.
• I. M. Vinogradov, Representation of an odd number as a sum of three primes, Comptes Rendus (Doklady) de l'Académie des Sciences de l'URSS 15 (1937), 291--294.
• Van Vu, High order complementary bases of primes, Integers 2 (2002).
• E. Wirsing, Thin subbases, Analysis 6 (1986), 285--308.