Abstract
We explore some of the numerical implications of the famous 1922 paper “Some Problems of ‘Partitio Numerorum’ III: On the Expression of a Number as a Sum of Primes” of Hardy and Littlewood. In particular, we prove that if the Generalized Riemann Hypothesis holds, then Hardy and Littlewood's analysis yields that every odd number greater than 1.24 × 1050 is a sum of three primes.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
E. Artin, The Gamma Function, Holt, Rinehart, Winston, 1964.
K.G. Borodzkin, "K voprosu o postoyanni I.M. Vinogradov," Trudy Tretego Vsesoiuznogo Matematiceskogo Siezda, Moskva, Vol. 1, 1956.
J. Chen and T. Wang, "On the odd Goldbach problem." Acta Math.Sinica 32(1989), 702–718.
J.-M. Deshouillers, G. Effinger, H. te Riele, and D. Zinoviev, "A complete Vinogradov 3-primes theorem under the Riemann hypothesis," Electronic Research Announcements of the American Mathematical Society 3(1997), 99–104.
G.W. Effinger and D.R. Hayes, Additive Number Theory of Polynomials over a Finite Field, Oxford University Press, New York, 1991.
G.W. Effinger and D.R. Hayes, "A complete solution to the polynomial 3-primes problem," Bulletin of the American Mathematical Society 24(1991), 363–369.
G.H. Hardy and J.E. Littlewood, "Some problems of 'partitio numerorum'; III: On the expression of a number as a sum of primes," Acta Mathematica 44(1922), 1–70.
C.G. van der Laan and N.M. Temme, Calculation of Special Functions: The Gamma Function, Exponential Integrals, and Error-Like Functions, Centrum voor Wiskunde en Informica, Amsterdam, 1980.
E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Vol. 1 (B.G. Teubner, ed.) Leipzig, 1909.
E. Landau, " ¨ Uber die Riemannische Zetafunktion in der N¨ahevon D 1," Rendiconte del Circolo Mathematica di Palermo 50(1926), 423–427.
Bruno Lucke, "Zur Hardy-Littlewoodschen Behandlung des Goldbachschen Problems," Doctoral Dissertation, Gottingen, 1926.
S.J. Patterson, An Introduction to the Theory of the Riemann Zeta Function, Cambridge University Press, 1988.
P. Ribenboim, The Book of Prime Number Records, Springer-Verlag, 1988.
E.C. Titchmarsh, "A divisor problem," Rendiconte del Circolo Mathematica di Palermo 54(1930), 414–429.
I.M. Vinogradov, "Representation of an odd number as a sum of three primes," Comptes Rendues (Doklady) de l 'Academy des Sciences de l'USSR, 15(1937), 191–294.
D. Zinoviev, "On Vinogradov's constant in Goldbach's ternary problem," Journal of Number Theory 65(1997), 334–358.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Effinger, G. Some Numerical Implications of the Hardy and Littlewood Analysis of the 3-Primes Problem. The Ramanujan Journal 3, 239–280 (1999). https://doi.org/10.1023/A:1009821519507
Issue Date:
DOI: https://doi.org/10.1023/A:1009821519507