Goldbach numbers in sparse sequences

Jörg Brüdern; Alberto Perelli

Annales de l'institut Fourier (1998)

  • Volume: 48, Issue: 2, page 353-378
  • ISSN: 0373-0956

Abstract

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We show that for almost all n ∈ N , the inequality | p 1 + p 2 - exp ( ( log n ) γ ) | < 1 has solutions with odd prime numbers p 1 and p 2 , provided 1 < γ < 3 2 . Moreover, we give a rather sharp bound for the exceptional set.This result provides almost-all results for Goldbach numbers in sequences rather thinner than the values taken by any polynomial.

How to cite

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Brüdern, Jörg, and Perelli, Alberto. "Goldbach numbers in sparse sequences." Annales de l'institut Fourier 48.2 (1998): 353-378. <http://eudml.org/doc/75285>.

@article{Brüdern1998,
abstract = {We show that for almost all $n\in \{\bf N\}$, the inequality\begin\{\}\vert p\_1+p\_2-\{\rm exp\}((\{\rm log\}\,n)^\gamma )\vert &lt; 1\end\{\}has solutions with odd prime numbers $p_1$ and $p_2$, provided $1&lt; \gamma &lt; \{3\over 2\}$. Moreover, we give a rather sharp bound for the exceptional set.This result provides almost-all results for Goldbach numbers in sequences rather thinner than the values taken by any polynomial.},
author = {Brüdern, Jörg, Perelli, Alberto},
journal = {Annales de l'institut Fourier},
keywords = {sparse sequence of even numbers; exponential sums; Goldbach numbers; Hardy-Littlewood method; Goldbach's problem},
language = {eng},
number = {2},
pages = {353-378},
publisher = {Association des Annales de l'Institut Fourier},
title = {Goldbach numbers in sparse sequences},
url = {http://eudml.org/doc/75285},
volume = {48},
year = {1998},
}

TY - JOUR
AU - Brüdern, Jörg
AU - Perelli, Alberto
TI - Goldbach numbers in sparse sequences
JO - Annales de l'institut Fourier
PY - 1998
PB - Association des Annales de l'Institut Fourier
VL - 48
IS - 2
SP - 353
EP - 378
AB - We show that for almost all $n\in {\bf N}$, the inequality\begin{}\vert p_1+p_2-{\rm exp}(({\rm log}\,n)^\gamma )\vert &lt; 1\end{}has solutions with odd prime numbers $p_1$ and $p_2$, provided $1&lt; \gamma &lt; {3\over 2}$. Moreover, we give a rather sharp bound for the exceptional set.This result provides almost-all results for Goldbach numbers in sequences rather thinner than the values taken by any polynomial.
LA - eng
KW - sparse sequence of even numbers; exponential sums; Goldbach numbers; Hardy-Littlewood method; Goldbach's problem
UR - http://eudml.org/doc/75285
ER -

References

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  1. [BHP] R.C. BAKER, G. HARMAN, J. PINTZ, The exceptional set for Goldbach's problem in short intervals - Sieve Methods, Exp. Sums and Appl. in Number Theory, ed. by G.R.H. Greaves et al., 1-54, Cambridge U. P. 1997. Zbl0929.11042MR99g:11121
  2. [BP] J. BRÜDERN, A. PERELLI, Goldbach numbers and uniform distribution mod 1, Analytic Number Theory, ed. by Y. Motohashi, 43-51, Cambridge Univ. Press. 1997. Zbl0911.11047
  3. [D] H. DAVENPORT, Multiplicative Number Theory, 2nd ed., Springer Verlag, 1980. Zbl0453.10002MR82m:10001
  4. [G] D.A. GOLDSTON, On Hardy and Littlewood's contribution to the Goldbach conjecture, Proc. Amalfi Conf. Analytic Number Theory, ed. by E. Bombieri et al., 115-155, Università di Salerno 1992. Zbl0792.11039MR94m:11122
  5. [HL] G.H. HARDY, J.E. LITTLEWOOD, Some problems of "Partitio Numerorum", V : A further contribution to the study of Goldbach's problem, Proc. London Math. Soc., (2) 22 (1923), 46-56. Zbl49.0127.03JFM49.0127.03
  6. [K] A.A. KARACUBA, Estimates for trigonometric sums by Vinogradov's method, and some applications, Proc. Steklov Inst. Math., 112 (1973), 251-265. Zbl0259.10040MR49 #7049
  7. [LP] A. LANGUASCO, A. PERELLI, A pair correlation hypothesis and the exceptional set in Goldbach's problem, Mathematika, 43 (1996), 349-361. Zbl0884.11042MR98g:11113
  8. [MV] H.L. MONTGOMERY, R.C. VAUGHAN, The exceptional set in Goldbach's problem, Acta Arith., 27 (1975), 353-370. Zbl0301.10043MR51 #10263
  9. [P] A. PERELLI, Goldbach numbers represented by polynomials, Rev. Math. Iberoamericana, 12 (1996), 477-490. Zbl0874.11069MR97h:11116
  10. [Va] R.C. VAUGHAN, The Hardy-Littlewood Method, Cambridge Univ. Press., 1981. Zbl0455.10034MR84b:10002
  11. [Vi] I.M. VINOGRADOV, Selected Works, Springer Verlag, 1985. 

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