Skip to main content
SpringerLink
Log in

Effective estimation of some oscillatory integrals related to infinitely divisible distributions

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

We present a practical framework to prove, in a simple way, two-term asymptotic expansions for Fourier integrals

$$\begin{aligned} {{\mathcal {I}}}(t) = \int _{{\mathbb {R}}}(\mathrm{e}^{it\phi (x)}-1) \mathop {}\!\mathrm {d}\mu (x), \end{aligned}$$

where \(\mu \) is a probability measure on \({{\mathbb {R}}}\) and \(\phi \) is measurable. This applies to many basic cases, in link with Levy’s continuity theorem. We present applications to limit laws related to rational continued fraction coefficients.

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. Bettin, S.: On the reciprocity law for the twisted second moment of Dirichlet L-functions. Trans. Am. Math. Soc. 368(10), 6887–6914 (2016)

    Article  MathSciNet  Google Scholar 

  2. Bettin, S., Drappeau, S.: Limit laws for rational continued fractions and value distribution of quantum modular forms (preprint)

  3. Bleistein, N., Handelsman, R.A.: Asymptotic Expansions of Integrals, 2nd edn, p. 863284. Dover, New York (1986)

    Google Scholar 

  4. Feller, W.: An Introduction to Probability Theory and Its Applications, vol. 2, 2nd edn, p. 0270403. Wiley, New York (1971)

    Google Scholar 

  5. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 7th edn. Elsevier, Amsterdam (2007), Translated from the Russian

  6. Ibragimov, I.A., Linnik, Yu.V.: Independent and stationary sequences of random variables. Wolters-Noordhoff Publishing, Groningen (1971), With a supplementary chapter by I. A. Ibragimov and V. V. Petrov, Translation from the Russian edited by J. F. C. Kingman, 0322926

  7. López, J.L.: Asymptotic expansions of Mellin convolution integrals. SIAM Rev. 50(2), 275–293 (2008)

    Article  MathSciNet  Google Scholar 

  8. Olver, F.W.J.: Asymptotics and special functions. AKP Classics, AK Peters, Ltd., Wellesley (1997), Reprint of the 1974 original [Academic Press, New York; MR0435697 (55 #8655)], 1429619

  9. Tenenbaum, G.: Introduction to Analytic and Probabilistic Number Theory. Graduate Studies in Mathematics, vol. 163, 3rd edn. American Mathematical Society, Providence (2015)

    Book  Google Scholar 

  10. Vardi, I.: Dedekind sums have a limiting distribution. Int. Math. Res. Notices 1, 1–12 (1993)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors thank the referee for his or her remarks and a careful reading of the paper. This paper was partially written during a visit of S. Bettin at the Aix-Marseille University and a visit of S. Drappeau at the University of Genova. The authors thank both Institution for the hospitality and Aix-Marseille University and INdAM for the financial support for these visits.

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Genova, via Dodecaneso 35, 16146, Genoa, Italy

    S. Bettin

  2. Aix Marseille Université, CNRS, Centrale Marseille, I2M UMR 7373, 13453, Marseille, France

    S. Drappeau

Authors
  1. S. Bettin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Drappeau
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Drappeau.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

S. Bettin is a member of the INdAM group GNAMPA and his work is partially supported by PRIN 2017 “Geometric, algebraic and analytic methods in arithmetic”. This work has benefitted from support by Aix-Marseille Université through Fonds d’Intervention Recherche Invités.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bettin, S., Drappeau, S. Effective estimation of some oscillatory integrals related to infinitely divisible distributions. Ramanujan J 57, 849–861 (2022). https://doi.org/10.1007/s11139-020-00362-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-020-00362-y

Keywords

Mathematics Subject Classification

Search

Navigation