Hostname: page-component-7479d7b7d-rvbq7 Total loading time: 0 Render date: 2024-07-13T08:00:08.287Z Has data issue: false hasContentIssue false
  • English
  • Français

On a theorem of Bombieri–Vinogradov type

Published online by Cambridge University Press:  26 February 2010

E. Fouvry  and
H. Iwaniec
E. Fouvry
Affiliation:
U.E.R. de Mathématiques et d'Informatique, Université de Bordeaux I, 351, Cours de la Libération, 33405 Talence, France
H. Iwaniec
Affiliation:
U.E.R. de Mathématiques et d'Informatique, Université de Bordeaux I, 351, Cours de la Libération, 33405 Talence, France
Get access

Extract

The celebrated theorem of Bombieri and A. I. Vinogradov states that

for any ε > 0 and A ε 0, the implied constant in the symbol «ε depending at most on ε and A (see [1] and [14]). The original proofs of Bombieri and Vinogradov were greatly simplified by P. X. Gallagher [4]. An elegant proof has been given recently by R. C. Vaughan [13]. For other references see H. L. Montgomery [10] and H. -E. Richert [12]. Estimates of type (1) are required in various applications of sieve methods. Having this in mind distinct generalizations have been investigated (see for example [15] and [2]). Y. Motohashi established a general theorem which, roughly speaking, says that if (1) holds for two arithmetic functions then it also holds for their Dirichlet convolution; for precise assumptions and statement see [11]. So far, all methods depend on the large sieve inequality (see [10])

which sets the limit x1/2 for the modulus q in (1) and in its generalizations.

Type
Research Article
Information
Mathematika , Volume 27 , Issue 2 , December 1980 , pp. 135 - 152
Copyright
Copyright © University College London 1980

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

1.Bombieri, E.. “On the large sieve”, Mathematika, 12 (1965), 201–225.CrossRefGoogle Scholar
2.Ding, X. and Pan, C. D.. “A new mean value theorem”, Sci. Sinica, Special Issue (II), (1979), 149–161.Google Scholar
3.Friedlander, J. and Iwaniec, H.. “Quadratic polynomials and quadratic forms”, Ada Math., 141 (1978), 1–15.Google Scholar
4.Gallagher, P. X.. “Bombieri's mean value theorem”, Mathematika, 15 (1968), 1–6.CrossRefGoogle Scholar
5.Halberstam, H. and Richert, H. -E.. Sieve Methods, (Academic Press, London-New York, 1974).Google Scholar
1.Hooley, C.. “On the number of divisors of quadratic polynomials” Ada Math., 110 (1963), 97–114.Google Scholar
7.Hooley, C.. “On the greatest prime factor of a cubic polynomial”, J. Reine angew. Math., 303/304 (1978), 21–50.Google Scholar
8.Iwaniec, H.. “Rosser's sieve”, Ada Arith., 36 (1978), 171–202.CrossRefGoogle Scholar
9.Kloosterman, H. D.. “On the representation of numbers in the form ax2 + by2 + cz2 + dt2”, Ada Math., 49 (1926), 407–464.Google Scholar
10.Montgomery, H. L.. Topics in Multiplicative Number Theory, Lecture Notes in Math. 227 (Berlin and New York, 1971).CrossRefGoogle Scholar
11.Motohashi, Y.. “An induction principle for the generalization of Bombieri's Prime Number Theorem”, Proc. Japan Acad., 52 (1976), 273–275.Google Scholar
12.Richert, H. -E.. Lectures on Sieve Methods, (Tata Inst. of Fund. Research, 1976).Google Scholar
13.Vaughan, R. C.. “On the estimation of trigonometric sums over primes and related questions”, Institut Mittag-Uffler Report No. 9 (1977).Google Scholar
14.Vinogradov, A. I.. “The density hypothesis for Dirichlet's L-series” (Russian), Izv. Akad. Nauk SSSR, Ser. Math. 29 (1965), 903–934; Corrigendum:Google Scholar
14.Vinogradov, A. I.. “The density hypothesis for Dirichlet's L-series” (Russian), Izv. Akad. Nauk SSSR, Ser. Math. 30 (1966), 719–720.Google Scholar
15.Wolke, D.. “Über die mittlere Verteilung der Werte zahlentheoretischen Funktionen auf Restklassen I, Math. Annul, 202 (1973), 1–25.CrossRefGoogle Scholar