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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A Wiener type theorem for Dirichlet series


Authors: Arthur Goodman and D. J. Newman
Journal: Proc. Amer. Math. Soc. 92 (1984), 521-527
MSC: Primary 30B50; Secondary 32A99, 46H99
MathSciNet review: 760938
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Abstract: A famous theorem of Wiener states the conditions under which the reciprocal of a function with an absolutely convergent Fourier series also has an absolutely convergent Fourier series.

We offer an elementary proof of the fact, first proven in [2], that if $ F(s)$ has an absolutely convergent Dirichlet series then $ 1/F(s)$ has an absolutely convergent Dirichlet series if and only if $ \left\vert {F(s)} \right\vert$ is bounded away from zero in the closed right half-plane.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1984-0760938-3
PII: S 0002-9939(1984)0760938-3
Keywords: Dirichlet series, Banach algebras, functions of several complex variables, positive basis, Kronecker's theorem, Reinhardt domain
Article copyright: © Copyright 1984 American Mathematical Society