manuscripta mathematica

, Volume 117, Issue 2, pp 183-197

First online:

Irrationality of Power Series for Various Number Theoretic Functions

  • William D. BanksAffiliated withDepartment of Mathematics, University of Missouri  Email author 
  • , Florian LucaAffiliated withInstituto de Matemáticas, Universidad Nacional Autónoma de México
  • , Igor E. ShparlinskiAffiliated withDepartment of Computing, Macquarie University

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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.