Natural Boundaries of Power Series with Multiplicative Coefficients in Algebraic Number Fields

Friedemann Tuttas ()
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Friedemann Tuttas: Hasengartenstrasse 13 B

A chapter in From Arithmetic to Zeta-Functions, 2016, pp 509-521 from Springer

Abstract: Abstract For an algebraic number field $$K\neq \mathbb{Q}$$ we prove that the unit disc is a natural boundary of the power series $$\sum _{\mathfrak{a}\neq \mathfrak{o}}z^{N(\mathfrak{a})}$$ , $$\mathfrak{a}$$ running through the integral ideals of K and N denoting the norm function. As an application, we deduce the same result for power series $$\sum _{\mathfrak{a}\neq \mathfrak{o}}g(\mathfrak{a})\,z^{N(\mathfrak{a})}$$ with specific multiplicative coefficients $$g(\mathfrak{a})$$ thereby extending known results to algebraic number fields.

Keywords: Algebraic number fields; Analytic continuation; Multiplicative arithmetic functions; Power series; Primary 11N37; Secondary 30B40 (search for similar items in EconPapers)
Date: 2016
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-28203-9_30

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DOI: 10.1007/978-3-319-28203-9_30

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