Illinois Journal of Mathematics

Differential transcendence of a class of generalized Dirichlet series

Masaaki Amou and Masanori Katsurada

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We investigate differential transcendence properties for a generalized Dirichlet series of the form $\sum_{n=0}^\infty a_n\lambda_n^{-s}$. Our treatment of this series is purely algebraic and does not rely on any analytic properties of generalized Dirichlet series. We establish differential transcendence theorems for a certain class of generalized Dirichlet series. These results imply that the Hurwits zeta-function $\zeta(s,a)$ does not satisfy an algebraic differential equation with complex coefficients.

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Illinois J. Math., Volume 45, Number 3 (2001), 939-948.

First available in Project Euclid: 13 November 2009

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Zentralblatt MATH identifier

Primary: 11J91: Transcendence theory of other special functions
Secondary: 11M35: Hurwitz and Lerch zeta functions 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72}


Amou, Masaaki; Katsurada, Masanori. Differential transcendence of a class of generalized Dirichlet series. Illinois J. Math. 45 (2001), no. 3, 939--948. doi:10.1215/ijm/1258138161.

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