Mathematical Research Letters

Volume 20 (2013)

Number 3

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

Pages: 429 – 448

DOI: https://dx.doi.org/10.4310/MRL.2013.v20.n3.a2

Authors

Bruce C. Berndt (Department of Mathematics, University of Illinois, Urbana, Il., U.S.A.)

Sun Kim (Department of Mathematics, Ohio State University, Columbus, Oh., U.S.A.)

Alexandru Zaharescu (Department of Mathematics, University of Illinois, Urbana, Il., U.S.A.; Institute of Mathematics, Romanian Academy, Bucharest, Romania)

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.

Keywords

diophantine approximation, diophantine inequalities, hypergeometric functions, Dirichlet $L$-functions, $L$-functions for elliptic curves, partial Taylor series sums

2010 Mathematics Subject Classification

Primary 11J70. Secondary 11J25, 11M06, 33C20.

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Published 9 January 2014