Mathematische Zeitschrift

, Volume 292, Issue 1–2, pp 499–511 | Cite as

Sondow’s conjecture, convergents to e, and p-adic analytic functions

  • Vicenţiu PaşolEmail author
  • Alexandru Zaharescu


In their study of diophantine approximation of the exponential function in connection with Sondow’s Conjecture, Berndt et al. (Adv Math 348:1298–1331, 2013) have constructed certain p-adic functions arising from the sequence of convergents to the continued fraction of e. We solve an open problem posed in [2], more precisely we show that those p-adic functions are locally analytic (of minimal radius 1 / 2). We leave open the question of the existence of nontrivial zeros (i.e. zeros that are not forced by the functional equations) for these functions.


Sondow’s Conjecture Continued fractions p-adic analytic functions 



  1. 1.
    Amice, Y.: Interpolation \(p\)-adique. Bull. de la Soc. Math. de Fr. 92, 117–180 (1964). (French)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Berndt, B.C., Kim, S., Zaharescu, A.: Diophantine approximation of the exponential function and Sondow’s Conjecture. Adv. Math. 248, 1298–1331 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Berndt, B.C., Kim, S., Zaharescu, A.: Dirichlet L-functions, elliptic curves, hypergeometric functions, and rational approximation with partial sums of power series. Math. Res. Lett. 20(3), 429–448 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Berndt, B.C., Kim, S., Phaovibul, M.T., Zaharescu, A.: Diophantine approximation with partial sums of power series. Acta Arith. 161(3), 249–266 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Koblitz, N.: \(p\)-adic Numbers, \(p\)-adic analysis, and zeta-functions, Graduate Texts in Mathematics, vol. 58, 2nd edn. Springer-Verlag, New York (1984). CrossRefGoogle Scholar
  6. 6.
    Mahler, K.: An interpolation series for continuous functions of a \(p\)-adic variable. J. Fr Die Reine und Angewandte Mathematik 199, 23–34 (1958)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Pollack, R., Stevens, G.: Overconvergent modular symbols and \(p\)-adic \(L\)-functions. Annales scientifiques de l’École Normale Supérieure, Série 4, 44(1), 1–42 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Sondow, J.: A geometric proof that \(e\) is irrational and a new measure of its irrationality. Am. Math. Mon. No. 113, 637–641 (2006)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Schalm, K., Sondow, J.: Which partial sums of the Taylor series for \(e\) are convergents to \(e\)? (and a link to the primes 2, 5, 13, 37, 463), part II. In: Amdeberhan, T., Medina, L.A., Moll, V.H. (eds.) Gems in Experimental Mathematics, Contemp. Math., vol. 517, pp. 349–363. American Mathematical Society, Providence, RI (2010)CrossRefGoogle Scholar
  10. 10.
    Stevens, G.: Coleman’s \(L\)-invariant and families of modular forms. Aster. No. 331, 1–12 (2010)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Strassmann, R.: Über den Wertevorrat von Potenzreihen im Gebiet der \(p\)-adischen Zahlen. J. Für die Reine und Angewandte Math. 159, 13–28 (1928). (In German)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Simion Stoilow Institute of Mathematics of the Romanian AcademyBucharestRomania
  2. 2.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA

Personalised recommendations