manuscripta mathematica

, Volume 117, Issue 2, pp 183–197

Irrationality of Power Series for Various Number Theoretic Functions

  • William D. Banks
  • Florian Luca
  • Igor E. Shparlinski
Article

Abstract

We study formal power series whose coefficients are taken to be a variety of number theoretic functions, such as the Euler, Möbius and divisor functions. We show that these power series are irrational over ℤ[X], and we obtain lower bounds on the precision of their rational approximations.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Allouche, J.-P.: ‘Transcendence of formal power series with rational coefficients’. Theoret. Comput. Sci. 218, 143–160 (1999)CrossRefGoogle Scholar
  2. 2.
    Erdős, P., Granville, A., Pomerance, C., Spiro, C.: ‘On the normal behaviour of the iterates of some arithmetic functions’. Analytic Number Theory, Birkhäuser, Boston, 1990, pp. 165–204Google Scholar
  3. 3.
    Everest, G., van der Poorten, A. J., Shparlinski, I. E., Ward, T. B.: Recurrence sequences. Amer. Math. Soc., 2003Google Scholar
  4. 4.
    Heath-Brown, R.: ‘Zero-free regions for Dirichlet L-functions and the least prime in an arithmetic progression’. Proc. Lond. Math. Soc. 64, 265–338 (1991)Google Scholar
  5. 5.
    Heath-Brown, R.: ‘Almost-primes in arithmetic progressions and short intervals’. Math. Proc. Cambridge Philos. Soc. 83, 357–375 (1978)Google Scholar
  6. 6.
    Littlewood, J. E.: ‘Sur la distribution des nombres premiers’. C. R. Acad. Sci. Paris 158, 1869–1872 (1914)Google Scholar
  7. 7.
    Lidl, R., Niederreiter, H.: Finite fields. Cambridge: Cambridge University Press, 1997Google Scholar
  8. 8.
    Luca, F., Pomerance, C.: ‘On some problems of Ma kowski–Schinzel and Erdős concerning the arithmetical functions ϕ and σ’. Colloq. Math. 92, 111–130 (2002)Google Scholar
  9. 9.
    Shparlinski, I. E.: Finite fields: Theory and computation. Kluwer Acad. Publ., Dordrecht, 1999Google Scholar
  10. 10.
    Suryanarayana, D., Sitaramachandra Rao, R.: ‘The distribution of square-full integers’. Arkiv för Matematik 11, 195–201 (1973)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • William D. Banks
    • 1
  • Florian Luca
    • 2
  • Igor E. Shparlinski
    • 3
  1. 1.Department of MathematicsUniversity of Missouri ColumbiaUSA
  2. 2.Instituto de MatemáticasUniversidad Nacional Autónoma de México MoreliaMéxico
  3. 3.Department of ComputingMacquarie UniversitySydneyAustralia

Personalised recommendations