Diophantine approximation of the exponential function and Sondow's Conjecture

Research output: Contribution to journalArticle

Abstract

We begin by examining a hitherto unexamined partial manuscript by Ramanujan on the diophantine approximation of e2/a published with his lost notebook. This diophantine approximation is then used to study the problem of how often the partial Taylor series sums of e coincide with the convergents of the (simple) continued fraction of e. We then develop a p-adic analysis of the denominators of the convergents of e and prove a conjecture of J. Sondow that there are only two instances when the convergents of the continued fraction of e coalesce with partial sums of e. We conclude with open questions about the zeros of certain p-adic functions naturally occurring in our proofs.

Original languageEnglish (US)
Pages (from-to)1298-1331
Number of pages34
JournalAdvances in Mathematics
Volume248
DOIs
StatePublished - Nov 25 2013

Keywords

  • Approximation by partial sums of power series
  • Continued fractions
  • Diophantine approximation
  • Exponential function
  • P-Adic analysis
  • Primary
  • Secondary
  • Sondow's Conjecture
  • Supercongruences

ASJC Scopus subject areas

  • Mathematics(all)

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