Dirichlet L-functions, elliptic curves, hypergeometric functions, and rational approximation with partial sums of power series

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Abstract

We consider the Diophantine approximation of exponential generating functions at rational arguments by their partial sums and by convergents of their (simple) continued fractions. We establish quantitative results showing that these two sets of approximations coincide very seldom. Moreover, we offer many conjectures about the frequency of their coalescence. In particular, we consider exponential generating functions with real Dirichlet characters and with coefficients of L-functions of elliptic curves, where calculational data provide striking examples showing agreement for certain convergents of high index and gargantuan heights. Finally, we similarly examine hypergeometric functions; note that e is a special case of the latter.

Original languageEnglish (US)
Pages (from-to)429-448
Number of pages20
JournalMathematical Research Letters
Volume20
Issue number3
DOIs
StatePublished - May 2013

Keywords

  • Diophantine approximation
  • Diophantine inequalities
  • Dirichlet L-functions
  • Hypergeometric functions
  • L-functions for elliptic curves
  • Partial Taylor series sums

ASJC Scopus subject areas

  • Mathematics(all)

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