Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-27T22:03:33.844Z Has data issue: false hasContentIssue false
  • English
  • Français

On the exceptional set for the sum of a prime and a k-th power

Part of: Additive number theory; partitions

Published online by Cambridge University Press:  26 February 2010

Alessandro Zaccagnini
Alessandro Zaccagnini
Affiliation:
Dipartimento di Matematica, Università di Genova, Via L. B. Alberti 4, 16132 Genova, Italy.
Get access

Abstract

Let k ≤ 2 be an integer, and set

Ek (X) = |{n ≤ X, n ≠ mk, n not a sum of a prime and a k-th power}|.

We prove that there exists δ = δ(k) > 0 such that Ek (X)Ek(X)≪kX1−δ.

MSC classification

Secondary: 11P32: Goldbach-type theorems; other additive questions involving primes
Type
Research Article
Information
Mathematika , Volume 39 , Issue 2 , December 1992 , pp. 400 - 421
Copyright
Copyright © University College London 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bo.Bombieri, E.. Le Grand Crible dans la Theorie Analytique des Nombres. Asterisque, n. 18, 1974.Google Scholar
BPP.Brünner, R., Perelli, A. and Pintz, J.. The exceptional set for the sum of a prime and a square. Ada Math. Hung., 53 (1989), 347–365.Google Scholar
Bu.Burgess, D. A.. On character sums and L-series. Proc. London Math. Soc. (3), 12 (1962), 193–206.Google Scholar
D.Davenport, H.. Multiplicative Number Theory, 2nd edtion (Springer Verlag, 1980).CrossRefGoogle Scholar
DH.Davenport, H. and Heilbronn, H.. Note on a result in the additive theory of numbers. Proc. London Math. Soc. (2), 43 (1937), 142–151.Google Scholar
G.Gallagher, P. X.. A large sieve density estimate near σ=l. Invent. Math., 11 (1970), 329–339.CrossRefGoogle Scholar
HL.Hardy, G. H. and Littlewood, J. E.. Some problems of “Partitio Numerorum”; III. On the expression of a number as a sum of primes. Acta Math., 44 (1923), 1–70.Google Scholar
M.Miech, R. J.. On the equation n = p + x2. Trans. Amer. Math. Soc, 130 (1968), 494–512.Google Scholar
MV1.Montgomery, H. L. and Vaughan, R. C.. Hilbert's inequality. J. London Math. Soc. (2), 8 (1974), 73–82.CrossRefGoogle Scholar
MV2.Montgomery, H. L. and Vaughan, R. C.. On the exceptional set in Goldbach's problem. Acta Arith., 27 (1975), 353–370.Google Scholar
Schm.Schmidt, W. M.. Equations over Finite Fields. An Elementary Approach, Lecture Notes No. 536 (Springer Verlag, 1976).Google Scholar
Schw.Schwarz, W.. Zur Darstellung von Zahlen durch summen von Primzahlpotenzen, II. Darstellungen fur “fast alle” Zahlen. J. reine angew. Math., 206 (1961), 78–112.Google Scholar
T.Titchmarsh, E. C.. The Theory of the Riemann Zeta-Function, 2nd edition (Oxford U.P., 1986).Google Scholar
Val.Vaughan, R. C.. On Goldbach's problem. Acta Arith., 22 (1972), 21–48.Google Scholar
Va2.Vaughan, R. C.. The Hardy-Littlewood Method (Cambridge U.P., 1981Google Scholar
Vi.Vinogradov, A. I.. On a binary problem of Hardy and Littlewood (Russian). Ada Arith., 46 (1985), 33–56.Google Scholar
W.Wilson, B. M.. Proofs of some formulae enunciated by Ramanujan. Proc. London Math. Soc. (2), 21 (1923), 235–255.Google Scholar