Hostname: page-component-74656f9884-8r8nv Total loading time: 0 Render date: 2026-11-10T09:33:48.207Z Has data issue: false hasContentIssue false
  • English
  • Français

THE MONTGOMERY–HOOLEY THEOREM IN SHORT INTERVALS

Part of: Additive number theory; partitions Multiplicative number theory

Published online by Cambridge University Press:  02 August 2012

Glyn Harman
Glyn Harman*
Affiliation:
Department of Mathematics, Royal Holloway, University of London, Egham Hill, Egham, Surrey TW20 0EX, U.K. (email: G.Harman@rhul.ac.uk)

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

In this paper we generalize and extend work by Languasco, Perreli and Zaccagnini on the Montgomery–Hooley theorem in short intervals. We are thus able to obtain asymptotic formulae for the mean-square error in the distribution of general sequences in arithmetic progressions in short intervals. As applications we consider lower and upper bound approximations to the characteristic function of the primes in a short interval and an average over short intervals for the von Mangoldt function.

MSC classification

Secondary: 11N13: Primes in progressions 11N36: Applications of sieve methods 11P55: Applications of the Hardy-Littlewood method

Information

Type
Research Article
Information
Mathematika , Volume 59 , Issue 1 , January 2013 , pp. 129 - 139
Copyright
Copyright © University College London 2012

References

[1]Baker, R. C., Harman, G. and Pintz, J., The exceptional set for Goldbach’s problem in short intervals. In Sieve Methods, Exponential Sums and Their Applications in Number Theory, Cambridge University Press (Cambridge, 1997), 11–54.Google Scholar
[2]Baker, R. C., Harman, G. and Pintz, J., The difference between consecutive primes II. Proc. Lond. Math. Soc. (3) 83 (2001), 532–562.Google Scholar
[3]Barban, M. B., Analogues of the divisor problem of Titchmarsh. Vestnik Leningrad Univ. Mat. Mekh. Astronom. 18 (1963), 5–13.Google Scholar
[4]Barban, M. B., The large sieve method and its applications in the theory of numbers. Uspekhi Mat. Nauk 21 (1966), 51–102.Google Scholar
[5]Brüdern, J. and Wooley, T. D., Sparse variance for primes in arithmetic progression. Q. J. Math. 62 (2011), 289–305.CrossRefGoogle Scholar
[6]Davenport, H., Multiplicative Number Theory, 3rd edn. (revised by H. L. Montgomery) (Graduate Texts in Mathematics 74), Springer (New York, NY, 2000).Google Scholar
[7]Davenport, H. and Halberstam, H., Primes in arithmetic progressions. Michigan Math. J. 13 (1968), 485–489.Google Scholar
[8]Gallagher, P. X., The large sieve. Mathematika 14 (1967), 14–20.CrossRefGoogle Scholar
[9]Goldston, D. A. and Vaughan, R. C., On the Montgomery–Hooley asymptotic formula. In Sieve Methods, Exponential Sums and Their Applications in Number Theory, Cambridge University Press (Cambridge, 1997), 11–54.Google Scholar
[10]Harman, G., Prime-detecting Sieves (London Mathematical Society Monographs Series 33), Princeton University Press (Princeton, NJ, 2007).Google Scholar
[11]Harman, G., Watt, N. and Wong, K. C., A new mean-value result for Dirichlet L-functions and polynomials. Q. J. Math. 55 (2004), 307–324.Google Scholar
[12]Heath-Brown, D. R., The differences between consecutive primes. J. Lond. Math. Soc. (2) 18 (1978), 7–13.CrossRefGoogle Scholar
[13]Heath-Brown, D. R., The differences between consecutive primes III. J. Lond. Math. Soc. (2) 20 (1979), 177–178.Google Scholar
[14]Hooley, C., On the Barban–Davenport Halberstam theorem III. J. Lond. Math. Soc. (2) 10 (1975), 249–256.Google Scholar
[15]Hooley, C., On the Barban–Davenport Halberstam theorem XIX. Hardy-Ramanujan J. 30 (2007), 56–67.Google Scholar
[16]Languasco, A., Perelli, A. and Zaccagnini, A., On the Montgomery–Hooley theorem in short intervals. Mathematika 56 (2010), 231–243.CrossRefGoogle Scholar
[17]Matomäki, K., Large differences between consecutive primes. Q. J. Math. 58 (2007), 489–517.CrossRefGoogle Scholar
[18]Montgomery, H. L., Primes in arithmetic progressions. Michigan Math. J. 17 (1970), 33–39.CrossRefGoogle Scholar
[19]Prachar, K., Primzahlverteilung, Springer (Berlin, 1957).Google Scholar
[20]Vaughan, R. C., The Hardy–Littlewood method, 2nd edn. (Cambridge Tracts in Mathematics, 80), Cambridge University Press (Cambridge, 1997).Google Scholar
[21]Vaughan, R. C., On a variance associated with the distribution of general sequences in arithmetic progressions I. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 356 (1998), 781–791.CrossRefGoogle Scholar
[22]Vaughan, R. C., On a variance associated with the distribution of primes in arithmetic progressions. Proc. Lond. Math. Soc. (3) 82 (2001), 533–553.CrossRefGoogle Scholar