Skip to main content
SpringerLink
Log in

Third power moments of the error terms corresponding to certain arithmetic functions

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

For positive integersn, letd(l 1,M 1;l 2,M 2;n) denote the number of factorizationsn=n 1 n 2 where each of the factorsn∈ℕ belongs to a prescribed congruence classl i moduloM i (i=1,2). In this article an asymptotic result is derived for the third power moment of the error term in the formula for the Dirichlet summmatory function ofd(l 1,M 1;l 2,M 2;n). This extends a recent result of [17] for the classic “unrestricted” case ofd(n)=d(1,1;1,1; n). Moreover, the technique developed is applied to the analogous problem related to Fourier coefficients of cusp forms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Apostol, T.: Introduction to analytic number theory. New York-Heidelberg-Berlin: Springer 1976

    MATH  Google Scholar 

  2. Chandrasekharan, K., andNarasimhan, R.: The approximate functional equation for a class of zeta-functions. Math. Ann.152, 30–64 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  3. Corrádi, K., andKátai, I.: A comment on K. S. Gangadharan’s paper “Two classical lattice point problems”. Magyar Tud. Akad. mat. fiz. Oszt. Közl.17, 89–97 (1967)

    MATH  Google Scholar 

  4. Deligne, P.: La conjecture de Weil. Inst. Hautes Etudes Sci. Publ. Math.53, 273–307 (1974)

    MathSciNet  Google Scholar 

  5. Hafner, J. L.: New omega theorems for two classical lattice point problems. Invent. math.63, 181–186 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  6. Heath-Brown, D.R.: The distribution and moments of the error term in the Dirichlet divisor problem. Acta Arithm.60, 389–415 (1992)

    MATH  MathSciNet  Google Scholar 

  7. Hlawka, E., Schoissengeier, J., andTaschner, R.: Geometric and analytic number theory. Berlin: Springer 1991

    MATH  Google Scholar 

  8. Huxley, M.N.: Exponential sums and lattice points, II. Proc. London Math. Soc. (3)66, 279–301 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  9. Ivić, A.: The Riemann zeta-function. New York-Chichester: J. Wiley & Sons 1985

    MATH  Google Scholar 

  10. Krätzel, E.: Lattice points. Dordrecht-Boston-London: Kluwer 1988

    MATH  Google Scholar 

  11. Landau, E.: Über die Gitterpunkte in einem Kreise. (Vierte Mitteilung). Nachr. Königl. Ges. Wiss., math.-naturwiss. Kl1923, 58–65 (1923)

    Google Scholar 

  12. Nowak, W. G.: On a divisor problem in arithmetic progressions. J. Number Theory31, 174–182 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  13. Nowak, W. G.: On the Piltz divisor problem with congruence conditions, II. Abh. Math. Sem. Univ. Hamburg60, 153–163 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  14. Rankin, R.A.: Sums of cusp form coefficients. In: ‘Automorphic Forms and Analytic Number Theory’ (ed. R. Murty) Publ. Centre Recherches Math., Montréal 1990, pp. 115–121.

    Google Scholar 

  15. Richert, H.-E.: Ein Gitterpunktproblem. Math. Ann.125, 467–471 (1953)

    Article  MATH  MathSciNet  Google Scholar 

  16. Tong, K.-C.: On divisor problems II, III (Chinese, English summary). Acta Math. Sinica6, 139–152 and 515–541 (1956)

    MathSciNet  Google Scholar 

  17. Tsang, K.-M.: Higher-power moments of Δ(x),E(t) andP(x). Proc. London Math. Soc. (3)65, 65–82 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  18. Voronoï, G.: Sur une fonction transcendante et ses applications à la sommation de quelques séries. Ann. Sci. École Norm. Sup. (3)21, 207–267 and 459–533 (1904)

    Google Scholar 

  19. Walfisz, A.: Über die Koeffizientensummen einiger Modulformen. Math. Ann.108, 75–90 (1933)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Statistik, TU Graz, A-8010, Graz

    Wolfgang Müller

  2. Institut für Mathematik u. Ang. Stat., Universität für Bodenkultur, A-1180, Wien

    Werner Georg Nowak

Authors
  1. Wolfgang Müller
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Werner Georg Nowak
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

In memory of Professor Karl Prachar

This article is part of a research project supported by theAustrian Science Foundation (Nr. P 9298-PHY)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Müller, W., Nowak, W.G. Third power moments of the error terms corresponding to certain arithmetic functions. Manuscripta Math 87, 459–480 (1995). https://doi.org/10.1007/BF02570487

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02570487

Keywords

Search

Navigation